%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP132^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:48 AM UTC 2026
% Result : Theorem 2.62s 1.63s
% Output : Refutation 2.62s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 3
% Syntax : Number of formulae : 18 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 58 ( 27 equ; 0 cnn)
% Maximal formula atoms : 5 ( 3 avg)
% Number of connectives : 173 ( 19 ~; 4 |; 5 &; 136 @)
% ( 0 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% Number of symbols : 106 ( 103 usr; 6 con; 0-6 aty)
% Number of variables : 25 ( 11 ^; 14 !; 0 ?; 25 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
set_nat: $tType ).
thf(type_def_8,type,
set_set_nat: $tType ).
thf(func_def_0,type,
set_set_nat: $tType ).
thf(func_def_1,type,
set_nat: $tType ).
thf(func_def_2,type,
nat: $tType ).
thf(func_def_3,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(func_def_4,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_5,type,
one_one_nat: nat ).
thf(func_def_6,type,
times_times_nat: nat > nat > nat ).
thf(func_def_7,type,
zero_zero_nat: nat ).
thf(func_def_8,type,
groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_9,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_10,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(func_def_11,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_12,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_13,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_14,type,
ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_15,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_16,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_17,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_18,type,
set_or1086813439et_nat: set_nat > set_set_nat ).
thf(func_def_19,type,
member_nat: nat > set_nat > $o ).
thf(func_def_20,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_21,type,
k: nat ).
thf(func_def_22,type,
n: nat ).
thf(func_def_23,type,
p: nat > nat ).
thf(func_def_38,type,
sP0: nat > nat > nat > $o ).
thf(func_def_39,type,
sP1: nat > nat > nat > $o ).
thf(func_def_40,type,
sP2: nat > nat > nat > $o ).
thf(func_def_41,type,
sP3: set_nat > ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_42,type,
sP4: set_nat > ( nat > nat ) > set_nat > $o ).
thf(func_def_43,type,
sK5: ( nat > set_nat ) > nat ).
thf(func_def_44,type,
sK6: ( nat > set_nat ) > nat ).
thf(func_def_45,type,
sK7: ( set_nat > set_nat ) > set_nat ).
thf(func_def_46,type,
sK8: ( set_nat > set_nat ) > set_nat ).
thf(func_def_47,type,
sK9: ( set_nat > nat ) > set_nat ).
thf(func_def_48,type,
sK10: ( set_nat > nat ) > set_nat ).
thf(func_def_49,type,
sK11: ( nat > nat ) > nat ).
thf(func_def_50,type,
sK12: ( nat > nat ) > nat ).
thf(func_def_51,type,
sK13: set_nat > ( nat > nat ) > nat ).
thf(func_def_52,type,
sK14: set_nat > ( nat > nat ) > nat ).
thf(func_def_53,type,
sK15: ( set_nat > nat ) > set_nat ).
thf(func_def_54,type,
sK16: ( set_nat > nat ) > set_nat ).
thf(func_def_55,type,
sK17: ( nat > nat ) > set_nat > nat ).
thf(func_def_56,type,
sK18: ( set_nat > set_nat ) > set_nat ).
thf(func_def_57,type,
sK19: ( set_nat > set_nat ) > set_nat ).
thf(func_def_58,type,
sK20: ( nat > nat ) > nat ).
thf(func_def_59,type,
sK21: ( nat > nat ) > nat ).
thf(func_def_60,type,
sK22: set_nat > ( nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_61,type,
sK23: ( set_nat > set_nat ) > set_nat ).
thf(func_def_62,type,
sK24: ( set_nat > set_nat ) > set_nat ).
thf(func_def_63,type,
sK25: ( nat > $o ) > nat ).
thf(func_def_64,type,
sK26: ( nat > nat ) > nat ).
thf(func_def_65,type,
sK27: ( nat > nat ) > nat ).
thf(func_def_66,type,
sK28: ( nat > nat ) > set_nat > ( nat > nat ) > nat ).
thf(func_def_67,type,
sK29: set_nat > ( nat > nat ) > nat ).
thf(func_def_68,type,
sK30: ( nat > nat ) > nat ).
thf(func_def_69,type,
sK31: ( nat > nat ) > nat ).
thf(func_def_70,type,
sK32: ( nat > set_nat ) > nat ).
thf(func_def_71,type,
sK33: ( nat > set_nat ) > nat ).
thf(func_def_72,type,
sK34: ( nat > nat ) > nat > nat ).
thf(func_def_73,type,
sK35: ( nat > nat ) > set_nat > ( nat > nat ) > ( nat > nat ) > set_nat > ( nat > nat ) > nat ).
thf(func_def_74,type,
sK36: set_nat > ( nat > nat ) > ( nat > nat ) > set_nat > nat ).
thf(func_def_75,type,
sK37: ( nat > $o ) > nat ).
thf(func_def_76,type,
sK38: ( set_nat > nat ) > set_nat ).
thf(func_def_77,type,
sK39: ( set_nat > nat ) > set_nat ).
thf(func_def_78,type,
sK40: ( nat > nat > $o ) > nat ).
thf(func_def_79,type,
sK41: ( nat > nat > $o ) > nat ).
thf(func_def_80,type,
sK42: ( nat > nat > $o ) > nat ).
thf(func_def_81,type,
sK43: ( nat > nat > $o ) > nat ).
thf(func_def_82,type,
sK44: ( nat > nat > $o ) > nat ).
thf(func_def_83,type,
sK45: nat > nat ).
thf(func_def_84,type,
sK46: set_nat > ( nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_85,type,
sK47: set_nat > ( nat > nat ) > ( nat > nat ) > set_nat > ( nat > nat ) > nat ).
thf(func_def_86,type,
sK48: set_nat > set_nat > ( nat > nat ) > nat ).
thf(func_def_87,type,
sK49: nat > set_nat > set_nat > ( nat > nat ) > nat ).
thf(func_def_88,type,
sK50: ( set_nat > nat ) > set_nat ).
thf(func_def_89,type,
sK51: ( set_nat > nat ) > set_nat ).
thf(func_def_90,type,
sK52: ( nat > set_nat ) > nat ).
thf(func_def_91,type,
sK53: ( nat > set_nat ) > nat ).
thf(func_def_92,type,
sK54: nat > ( nat > $o ) > nat ).
thf(func_def_93,type,
sK55: ( nat > $o ) > nat ).
thf(func_def_94,type,
sK56: ( set_nat > set_nat ) > set_nat ).
thf(func_def_95,type,
sK57: ( set_nat > set_nat ) > set_nat ).
thf(func_def_96,type,
sK58: set_nat > ( nat > nat ) > set_nat > nat ).
thf(func_def_97,type,
sK59: set_nat > ( nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_98,type,
sK60: ( nat > nat ) > ( nat > nat ) > set_nat > nat ).
thf(func_def_99,type,
sK61: set_nat > ( nat > nat ) > set_nat > nat ).
thf(func_def_100,type,
sK62: ( nat > nat ) > ( nat > nat ) > ( nat > nat ) > set_nat > nat ).
thf(func_def_101,type,
sK63: set_nat > set_nat > nat ).
thf(func_def_102,type,
sK64: ( nat > nat ) > nat ).
thf(func_def_103,type,
sK65: ( nat > nat ) > nat ).
thf(func_def_104,type,
sK66: ( nat > $o ) > nat ).
thf(func_def_105,type,
sK67: ( nat > $o ) > nat ).
thf(func_def_106,type,
sK68: nat > ( nat > $o ) > nat ).
thf(func_def_107,type,
sK69: ( nat > nat ) > nat ).
thf(func_def_108,type,
sK70: ( nat > nat ) > nat ).
thf(func_def_109,type,
sK71: ( nat > set_nat ) > nat ).
thf(func_def_110,type,
sK72: ( nat > set_nat ) > nat ).
thf(func_def_111,type,
sK73: ( nat > nat > $o ) > nat ).
thf(func_def_112,type,
sK74: ( nat > nat > $o ) > nat ).
thf(func_def_113,type,
sK75: ( nat > nat > $o ) > nat ).
thf(func_def_114,type,
sK76: ( nat > nat > $o ) > nat ).
thf(func_def_115,type,
sK77: ( nat > nat ) > nat ).
thf(func_def_116,type,
sK78: ( nat > nat ) > nat ).
thf(func_def_117,type,
sK79: ( nat > nat ) > nat ).
thf(func_def_118,type,
sK80: ( nat > nat ) > nat ).
thf(func_def_120,type,
ph82:
!>[X0: $tType] : X0 ).
thf(f1539,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1529,f1156]) ).
thf(f1156,plain,
( ( number1551313001itions @ p @ n )
= $true ),
inference(cnf_transformation,[status(thm)],[f380]) ).
thf(f380,plain,
( ( number1551313001itions @ p @ n )
= $true ),
inference(fool_elimination,[status(thm)],[f379]) ).
thf(f379,plain,
number1551313001itions @ p @ n,
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
number1551313001itions @ p @ n,
file('/export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2THF_2249.p',fact_0_partitions) ).
thf(f1529,plain,
( ( number1551313001itions @ p @ n )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1522]) ).
thf(f1522,plain,
( ( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat @ n @ k ) )
| ( ( number1551313001itions @ p @ n )
!= $true ) ),
inference(superposition,[status(thm)],[f1314,f1387]) ).
thf(f1387,plain,
! [X0: nat > nat,X1: nat] :
( ( ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( X0 @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ X1 ) )
= X1 )
| ( ( number1551313001itions @ X0 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f798]) ).
thf(f798,plain,
! [X0: nat > nat,X1: nat] :
( ( ( number1551313001itions @ X0 @ X1 )
!= $true )
| ( ( ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( X0 @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ X1 ) )
= X1 )
& ! [X2: nat] :
( ( zero_zero_nat
= ( X0 @ X2 ) )
| ( ( ( ord_less_eq_nat @ X2 @ X1 )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X2 )
= $true ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f618]) ).
thf(f618,plain,
! [X0: nat > nat,X1: nat] :
( ( ( number1551313001itions @ X0 @ X1 )
= $true )
=> ~ ( ! [X2: nat] :
( ( zero_zero_nat
!= ( X0 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ X1 )
= $true )
& ( ( ord_less_eq_nat @ one_one_nat @ X2 )
= $true ) ) )
=> ( ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( X0 @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ X1 ) )
!= X1 ) ) ),
inference(fool_elimination,[status(thm)],[f617]) ).
thf(f617,plain,
! [X0: nat > nat,X1: nat] :
( ( number1551313001itions @ X0 @ X1 )
=> ~ ( ! [X2: nat] :
( ( zero_zero_nat
!= ( X0 @ X2 ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ X2 )
& ( ord_less_eq_nat @ X2 @ X1 ) ) )
=> ( ( groups1842438620at_nat
@ ^ [X3: nat] : ( times_times_nat @ ( X0 @ X3 ) @ X3 )
@ ( set_ord_atMost_nat @ X1 ) )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f64]) ).
thf(f64,axiom,
! [X23: nat > nat,X11: nat] :
( ( number1551313001itions @ X23 @ X11 )
=> ~ ( ! [X45: nat] :
( ( zero_zero_nat
!= ( X23 @ X45 ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ X45 )
& ( ord_less_eq_nat @ X45 @ X11 ) ) )
=> ( ( groups1842438620at_nat
@ ^ [X2: nat] : ( times_times_nat @ ( X23 @ X2 ) @ X2 )
@ ( set_ord_atMost_nat @ X11 ) )
!= X11 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2THF_2249.p',fact_63_partitionsE) ).
thf(f1314,plain,
( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat
@ ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ n ) )
@ k ) ),
inference(cnf_transformation,[status(thm)],[f644]) ).
thf(f644,plain,
( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat
@ ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ n ) )
@ k ) ),
inference(flattening,[status(thm)],[f309]) ).
thf(f309,plain,
( ( minus_minus_nat @ n @ k )
!= ( minus_minus_nat
@ ( groups1842438620at_nat
@ ^ [Y0: nat] : ( times_times_nat @ ( p @ Y0 ) @ Y0 )
@ ( set_ord_atMost_nat @ n ) )
@ k ) ),
inference(fool_elimination,[status(thm)],[f308]) ).
thf(f308,plain,
( ( minus_minus_nat
@ ( groups1842438620at_nat
@ ^ [X0: nat] : ( times_times_nat @ ( p @ X0 ) @ X0 )
@ ( set_ord_atMost_nat @ n ) )
@ k )
!= ( minus_minus_nat @ n @ k ) ),
inference(rectify,[status(thm)],[f246]) ).
thf(f246,negated_conjecture,
( ( minus_minus_nat
@ ( groups1842438620at_nat
@ ^ [X2: nat] : ( times_times_nat @ ( p @ X2 ) @ X2 )
@ ( set_ord_atMost_nat @ n ) )
@ k )
!= ( minus_minus_nat @ n @ k ) ),
inference(negated_conjecture,[status(cth)],[f245]) ).
thf(f245,conjecture,
( ( minus_minus_nat
@ ( groups1842438620at_nat
@ ^ [X2: nat] : ( times_times_nat @ ( p @ X2 ) @ X2 )
@ ( set_ord_atMost_nat @ n ) )
@ k )
= ( minus_minus_nat @ n @ k ) ),
file('/export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2THF_2249.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP132^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n011.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:17 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP132^1 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer Number_Partition problem prob_191__5326608_1
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des21]
% 0.30/0.46 % Names : Number_Partition/prob_191__5326608_1 [Des21]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.45 v7.5.0
% 0.30/0.46 % Syntax : Number of formulae : 269 ( 120 unt; 24 typ; 0 def)
% 0.30/0.46 % Number of atoms : 672 ( 258 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.30/0.46 % Number of connectives : 1911 ( 62 ~; 17 |; 29 &;1502 @)
% 0.30/0.46 % ( 0 <=>; 301 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 17 ( 6 avg)
% 0.30/0.46 % Number of types : 4 ( 3 usr)
% 0.30/0.46 % Number of type conns : 116 ( 116 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 24 ( 21 usr; 6 con; 0-4 aty)
% 0.30/0.46 % Number of variables : 675 ( 63 ^; 606 !; 6 ?; 675 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:43:17.279
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (3)
% 0.30/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.30/0.46 set_set_nat: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.46 set_nat: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.30/0.46 nat: $tType ).
% 0.30/0.46
% 0.30/0.46 % Explicit typings (21)
% 0.30/0.46 thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.46 fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.30/0.46 minus_minus_nat: nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.30/0.46 one_one_nat: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 0.30/0.46 times_times_nat: nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.30/0.46 zero_zero_nat: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.46 groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If_001t__Nat__Onat,type,
% 0.30/0.46 if_nat: $o > nat > nat > nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 0.30/0.46 number1551313001itions: ( nat > nat ) > nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.30/0.46 ord_less_nat: nat > nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.30/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.46 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.30/0.46 ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.30/0.46 collect_nat: ( nat > $o ) > set_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.46 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 0.30/0.46 set_ord_atMost_nat: nat > set_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.46 set_or1086813439et_nat: set_nat > set_set_nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member_001t__Nat__Onat,type,
% 0.30/0.46 member_nat: nat > set_nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.46 member_set_nat: set_nat > set_set_nat > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_k,type,
% 0.30/0.46 k: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_n,type,
% 0.30/0.46 n: nat ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_p,type,
% 0.30/0.46 p: nat > nat ).
% 0.30/0.46
% 0.30/0.46 % Relevant facts (241)
% 0.30/0.46 thf(fact_0_partitions,axiom,
% 0.30/0.46 number1551313001itions @ p @ n ).
% 0.30/0.46
% 0.30/0.46 % partitions
% 0.30/0.46 thf(fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 0.30/0.46 ord_less_eq_nat @ k @ n ).
% 0.30/0.46
% 0.30/0.46 % \<open>k \<le> n\<close>
% 0.30/0.46 thf(fact_2_atMost__eq__iff,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ( ( set_ord_atMost_nat @ X )
% 0.30/0.46 = ( set_ord_atMost_nat @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_eq_iff
% 0.30/0.46 thf(fact_3__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062,axiom,
% 0.30/0.46 ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( fun_upd_nat_nat @ p @ k @ ( minus_minus_nat @ ( p @ k ) @ one_one_nat ) @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ ( minus_minus_nat @ n @ k ) ) )
% 0.30/0.46 = ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( fun_upd_nat_nat @ p @ k @ ( minus_minus_nat @ ( p @ k ) @ one_one_nat ) @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ n ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % \<open>(\<Sum>i\<le>n - k. (p(k := p k - 1)) i * i) = (\<Sum>i\<le>n. (p(k := p k - 1)) i * i)\<close>
% 0.30/0.46 thf(fact_4_calculation,axiom,
% 0.30/0.46 ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( fun_upd_nat_nat @ p @ k @ ( minus_minus_nat @ ( p @ k ) @ one_one_nat ) @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ ( minus_minus_nat @ n @ k ) ) )
% 0.30/0.46 = ( minus_minus_nat
% 0.30/0.46 @ ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( p @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ n ) )
% 0.30/0.46 @ k ) ) ).
% 0.30/0.46
% 0.30/0.46 % calculation
% 0.30/0.46 thf(fact_5_sum__product,axiom,
% 0.30/0.46 ! [F: nat > nat,A: set_nat,G: nat > nat,B: set_nat] :
% 0.30/0.46 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ B ) )
% 0.30/0.46 = ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] :
% 0.30/0.46 ( groups1842438620at_nat
% 0.30/0.46 @ ^ [J: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J ) )
% 0.30/0.46 @ B )
% 0.30/0.46 @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_product
% 0.30/0.46 thf(fact_6_sum__distrib__left,axiom,
% 0.30/0.46 ! [R: nat,F: nat > nat,A: set_nat] :
% 0.30/0.46 ( ( times_times_nat @ R @ ( groups1842438620at_nat @ F @ A ) )
% 0.30/0.46 = ( groups1842438620at_nat
% 0.30/0.46 @ ^ [N: nat] : ( times_times_nat @ R @ ( F @ N ) )
% 0.30/0.46 @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_distrib_left
% 0.30/0.46 thf(fact_7_sum__distrib__right,axiom,
% 0.30/0.46 ! [F: nat > nat,A: set_nat,R: nat] :
% 0.30/0.46 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ R )
% 0.30/0.46 = ( groups1842438620at_nat
% 0.30/0.46 @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R )
% 0.30/0.46 @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_distrib_right
% 0.30/0.46 thf(fact_8_diff__mult__distrib,axiom,
% 0.30/0.46 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.46 ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
% 0.30/0.46 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_mult_distrib
% 0.30/0.46 thf(fact_9_diff__mult__distrib2,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.46 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_mult_distrib2
% 0.30/0.46 thf(fact_10_atMost__subset__iff,axiom,
% 0.30/0.46 ! [X: set_nat,Y: set_nat] :
% 0.30/0.46 ( ( ord_le1613022364et_nat @ ( set_or1086813439et_nat @ X ) @ ( set_or1086813439et_nat @ Y ) )
% 0.30/0.46 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_subset_iff
% 0.30/0.46 thf(fact_11_atMost__subset__iff,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
% 0.30/0.46 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_subset_iff
% 0.30/0.46 thf(fact_12_atMost__iff,axiom,
% 0.30/0.46 ! [I2: set_nat,K: set_nat] :
% 0.30/0.46 ( ( member_set_nat @ I2 @ ( set_or1086813439et_nat @ K ) )
% 0.30/0.46 = ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_iff
% 0.30/0.46 thf(fact_13_atMost__iff,axiom,
% 0.30/0.46 ! [I2: nat,K: nat] :
% 0.30/0.46 ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
% 0.30/0.46 = ( ord_less_eq_nat @ I2 @ K ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_iff
% 0.30/0.46 thf(fact_14_diff__diff__cancel,axiom,
% 0.30/0.46 ! [I2: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ I2 @ N2 )
% 0.30/0.46 => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
% 0.30/0.46 = I2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_diff_cancel
% 0.30/0.46 thf(fact_15_nat__mult__eq__1__iff,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ M @ N2 )
% 0.30/0.46 = one_one_nat )
% 0.30/0.46 = ( ( M = one_one_nat )
% 0.30/0.46 & ( N2 = one_one_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mult_eq_1_iff
% 0.30/0.46 thf(fact_16_nat__1__eq__mult__iff,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( one_one_nat
% 0.30/0.46 = ( times_times_nat @ M @ N2 ) )
% 0.30/0.46 = ( ( M = one_one_nat )
% 0.30/0.46 & ( N2 = one_one_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_1_eq_mult_iff
% 0.30/0.46 thf(fact_17_le__refl,axiom,
% 0.30/0.46 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 0.30/0.46
% 0.30/0.46 % le_refl
% 0.30/0.46 thf(fact_18_le__trans,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ J2 @ K )
% 0.30/0.46 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_trans
% 0.30/0.46 thf(fact_19_eq__imp__le,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( M = N2 )
% 0.30/0.46 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_imp_le
% 0.30/0.46 thf(fact_20_le__antisym,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.46 => ( M = N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_antisym
% 0.30/0.46 thf(fact_21_nat__le__linear,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.46 | ( ord_less_eq_nat @ N2 @ M ) ) ).
% 0.30/0.46
% 0.30/0.46 % nat_le_linear
% 0.30/0.46 thf(fact_22_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.46 ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.46 ( ( P @ K )
% 0.30/0.46 => ( ! [Y2: nat] :
% 0.30/0.46 ( ( P @ Y2 )
% 0.30/0.46 => ( ord_less_eq_nat @ Y2 @ B2 ) )
% 0.30/0.46 => ? [X2: nat] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 & ! [Y3: nat] :
% 0.30/0.46 ( ( P @ Y3 )
% 0.30/0.46 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Nat.ex_has_greatest_nat
% 0.30/0.46 thf(fact_23_bounded__Max__nat,axiom,
% 0.30/0.46 ! [P: nat > $o,X: nat,M2: nat] :
% 0.30/0.46 ( ( P @ X )
% 0.30/0.46 => ( ! [X2: nat] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 => ( ord_less_eq_nat @ X2 @ M2 ) )
% 0.30/0.46 => ~ ! [M3: nat] :
% 0.30/0.46 ( ( P @ M3 )
% 0.30/0.46 => ~ ! [X3: nat] :
% 0.30/0.46 ( ( P @ X3 )
% 0.30/0.46 => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bounded_Max_nat
% 0.30/0.46 thf(fact_24_partitions__bounds,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat,I2: nat] :
% 0.30/0.46 ( ( number1551313001itions @ P2 @ N2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( P2 @ I2 ) @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_bounds
% 0.30/0.46 thf(fact_25_diff__le__mono2,axiom,
% 0.30/0.46 ! [M: nat,N2: nat,L: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_le_mono2
% 0.30/0.46 thf(fact_26_le__diff__iff_H,axiom,
% 0.30/0.46 ! [A2: nat,C: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ C )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
% 0.30/0.46 = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_diff_iff'
% 0.30/0.46 thf(fact_27_diff__le__self,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% 0.30/0.46
% 0.30/0.46 % diff_le_self
% 0.30/0.46 thf(fact_28_diff__le__mono,axiom,
% 0.30/0.46 ! [M: nat,N2: nat,L: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_le_mono
% 0.30/0.46 thf(fact_29_Nat_Odiff__diff__eq,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.46 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.46 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.46 = ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Nat.diff_diff_eq
% 0.30/0.46 thf(fact_30_le__diff__iff,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.46 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.46 = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_diff_iff
% 0.30/0.46 thf(fact_31_eq__diff__iff,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.46 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.46 => ( ( ( minus_minus_nat @ M @ K )
% 0.30/0.46 = ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.46 = ( M = N2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_diff_iff
% 0.30/0.46 thf(fact_32_mult__le__mono2,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_mono2
% 0.30/0.46 thf(fact_33_mult__le__mono1,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_mono1
% 0.30/0.46 thf(fact_34_mult__le__mono,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat,L: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_mono
% 0.30/0.46 thf(fact_35_le__square,axiom,
% 0.30/0.46 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_square
% 0.30/0.46 thf(fact_36_le__cube,axiom,
% 0.30/0.46 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cube
% 0.30/0.46 thf(fact_37_nat__mult__1__right,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( times_times_nat @ N2 @ one_one_nat )
% 0.30/0.46 = N2 ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mult_1_right
% 0.30/0.46 thf(fact_38_nat__mult__1,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( times_times_nat @ one_one_nat @ N2 )
% 0.30/0.46 = N2 ) ).
% 0.30/0.46
% 0.30/0.46 % nat_mult_1
% 0.30/0.46 thf(fact_39_sum__mono,axiom,
% 0.30/0.46 ! [K2: set_nat,F: nat > nat,G: nat > nat] :
% 0.30/0.46 ( ! [I3: nat] :
% 0.30/0.46 ( ( member_nat @ I3 @ K2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.46 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ K2 ) @ ( groups1842438620at_nat @ G @ K2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_mono
% 0.30/0.46 thf(fact_40_atMost__def,axiom,
% 0.30/0.46 ( set_or1086813439et_nat
% 0.30/0.46 = ( ^ [U: set_nat] :
% 0.30/0.46 ( collect_set_nat
% 0.30/0.46 @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_def
% 0.30/0.46 thf(fact_41_atMost__def,axiom,
% 0.30/0.46 ( set_ord_atMost_nat
% 0.30/0.46 = ( ^ [U: nat] :
% 0.30/0.46 ( collect_nat
% 0.30/0.46 @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % atMost_def
% 0.30/0.46 thf(fact_42_sum__subtractf__nat,axiom,
% 0.30/0.46 ! [A: set_nat,G: nat > nat,F: nat > nat] :
% 0.30/0.46 ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ A )
% 0.30/0.46 => ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
% 0.30/0.46 => ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.30/0.46 @ A )
% 0.30/0.46 = ( minus_minus_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_subtractf_nat
% 0.30/0.46 thf(fact_43_partitions__parts__bounded,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat] :
% 0.30/0.46 ( ( number1551313001itions @ P2 @ N2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ P2 @ ( set_ord_atMost_nat @ N2 ) ) @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_parts_bounded
% 0.30/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A2: nat,P: nat > $o] :
% 0.30/0.46 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.30/0.46 = ( P @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: set_nat] :
% 0.30/0.46 ( ( collect_nat
% 0.30/0.46 @ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_46_Collect__cong,axiom,
% 0.30/0.46 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.46 ( ! [X2: nat] :
% 0.30/0.46 ( ( P @ X2 )
% 0.30/0.46 = ( Q @ X2 ) )
% 0.30/0.46 => ( ( collect_nat @ P )
% 0.30/0.46 = ( collect_nat @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_47_sum_Oreindex__bij__witness,axiom,
% 0.30/0.46 ! [S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
% 0.30/0.46 ( ! [A3: nat] :
% 0.30/0.46 ( ( member_nat @ A3 @ S )
% 0.30/0.46 => ( ( I2 @ ( J2 @ A3 ) )
% 0.30/0.46 = A3 ) )
% 0.30/0.46 => ( ! [A3: nat] :
% 0.30/0.46 ( ( member_nat @ A3 @ S )
% 0.30/0.46 => ( member_nat @ ( J2 @ A3 ) @ T ) )
% 0.30/0.46 => ( ! [B3: nat] :
% 0.30/0.46 ( ( member_nat @ B3 @ T )
% 0.30/0.46 => ( ( J2 @ ( I2 @ B3 ) )
% 0.30/0.46 = B3 ) )
% 0.30/0.46 => ( ! [B3: nat] :
% 0.30/0.46 ( ( member_nat @ B3 @ T )
% 0.30/0.46 => ( member_nat @ ( I2 @ B3 ) @ S ) )
% 0.30/0.46 => ( ! [A3: nat] :
% 0.30/0.46 ( ( member_nat @ A3 @ S )
% 0.30/0.46 => ( ( H @ ( J2 @ A3 ) )
% 0.30/0.46 = ( G @ A3 ) ) )
% 0.30/0.46 => ( ( groups1842438620at_nat @ G @ S )
% 0.30/0.46 = ( groups1842438620at_nat @ H @ T ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.reindex_bij_witness
% 0.30/0.46 thf(fact_48_sum_Oeq__general__inverses,axiom,
% 0.30/0.46 ! [B: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 0.30/0.46 ( ! [Y2: nat] :
% 0.30/0.46 ( ( member_nat @ Y2 @ B )
% 0.30/0.46 => ( ( member_nat @ ( K @ Y2 ) @ A )
% 0.30/0.46 & ( ( H @ ( K @ Y2 ) )
% 0.30/0.46 = Y2 ) ) )
% 0.30/0.46 => ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ A )
% 0.30/0.46 => ( ( member_nat @ ( H @ X2 ) @ B )
% 0.30/0.46 & ( ( K @ ( H @ X2 ) )
% 0.30/0.46 = X2 )
% 0.30/0.46 & ( ( Gamma @ ( H @ X2 ) )
% 0.30/0.46 = ( Phi @ X2 ) ) ) )
% 0.30/0.46 => ( ( groups1842438620at_nat @ Phi @ A )
% 0.30/0.46 = ( groups1842438620at_nat @ Gamma @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.eq_general_inverses
% 0.30/0.46 thf(fact_49_sum_Oeq__general,axiom,
% 0.30/0.46 ! [B: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 0.30/0.46 ( ! [Y2: nat] :
% 0.30/0.46 ( ( member_nat @ Y2 @ B )
% 0.30/0.46 => ? [X3: nat] :
% 0.30/0.46 ( ( member_nat @ X3 @ A )
% 0.30/0.46 & ( ( H @ X3 )
% 0.30/0.46 = Y2 )
% 0.30/0.46 & ! [Ya: nat] :
% 0.30/0.46 ( ( ( member_nat @ Ya @ A )
% 0.30/0.46 & ( ( H @ Ya )
% 0.30/0.46 = Y2 ) )
% 0.30/0.46 => ( Ya = X3 ) ) ) )
% 0.30/0.46 => ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ A )
% 0.30/0.46 => ( ( member_nat @ ( H @ X2 ) @ B )
% 0.30/0.46 & ( ( Gamma @ ( H @ X2 ) )
% 0.30/0.46 = ( Phi @ X2 ) ) ) )
% 0.30/0.46 => ( ( groups1842438620at_nat @ Phi @ A )
% 0.30/0.46 = ( groups1842438620at_nat @ Gamma @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.eq_general
% 0.30/0.46 thf(fact_50_sum_Ocong,axiom,
% 0.30/0.46 ! [A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ B )
% 0.30/0.46 => ( ( G @ X2 )
% 0.30/0.46 = ( H @ X2 ) ) )
% 0.30/0.46 => ( ( groups1842438620at_nat @ G @ A )
% 0.30/0.46 = ( groups1842438620at_nat @ H @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.cong
% 0.30/0.46 thf(fact_51_diff__commute,axiom,
% 0.30/0.46 ! [I2: nat,J2: nat,K: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
% 0.30/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_commute
% 0.30/0.46 thf(fact_52_sum_Oswap,axiom,
% 0.30/0.46 ! [G: nat > nat > nat,B: set_nat,A: set_nat] :
% 0.30/0.46 ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( groups1842438620at_nat @ ( G @ I ) @ B )
% 0.30/0.46 @ A )
% 0.30/0.46 = ( groups1842438620at_nat
% 0.30/0.46 @ ^ [J: nat] :
% 0.30/0.46 ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( G @ I @ J )
% 0.30/0.46 @ A )
% 0.30/0.46 @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.swap
% 0.30/0.46 thf(fact_53_mult_Oright__neutral,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % mult.right_neutral
% 0.30/0.46 thf(fact_54_mult_Oleft__neutral,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % mult.left_neutral
% 0.30/0.46 thf(fact_55_fun__upd__upd,axiom,
% 0.30/0.46 ! [F: nat > nat,X: nat,Y: nat,Z: nat] :
% 0.30/0.46 ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ F @ X @ Y ) @ X @ Z )
% 0.30/0.46 = ( fun_upd_nat_nat @ F @ X @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_upd
% 0.30/0.46 thf(fact_56_fun__upd__triv,axiom,
% 0.30/0.46 ! [F: nat > nat,X: nat] :
% 0.30/0.46 ( ( fun_upd_nat_nat @ F @ X @ ( F @ X ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_triv
% 0.30/0.46 thf(fact_57_fun__upd__apply,axiom,
% 0.30/0.46 ( fun_upd_nat_nat
% 0.30/0.46 = ( ^ [F2: nat > nat,X4: nat,Y4: nat,Z2: nat] : ( if_nat @ ( Z2 = X4 ) @ Y4 @ ( F2 @ Z2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_upd_apply
% 0.30/0.46 thf(fact_58_order__refl,axiom,
% 0.30/0.46 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_59_order__refl,axiom,
% 0.30/0.46 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_60_lambda__one,axiom,
% 0.30/0.46 ( ( ^ [X4: nat] : X4 )
% 0.30/0.46 = ( times_times_nat @ one_one_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % lambda_one
% 0.30/0.46 thf(fact_61_partitions__def,axiom,
% 0.30/0.46 ( number1551313001itions
% 0.30/0.46 = ( ^ [P3: nat > nat,N: nat] :
% 0.30/0.46 ( ! [I: nat] :
% 0.30/0.46 ( ( ( P3 @ I )
% 0.30/0.46 != zero_zero_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ one_one_nat @ I )
% 0.30/0.46 & ( ord_less_eq_nat @ I @ N ) ) )
% 0.30/0.46 & ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( P3 @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.46 = N ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_def
% 0.30/0.46 thf(fact_62_partitionsI,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat] :
% 0.30/0.46 ( ! [I3: nat] :
% 0.30/0.46 ( ( ( P2 @ I3 )
% 0.30/0.46 != zero_zero_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ one_one_nat @ I3 )
% 0.30/0.46 & ( ord_less_eq_nat @ I3 @ N2 ) ) )
% 0.30/0.46 => ( ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( P2 @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ N2 ) )
% 0.30/0.46 = N2 )
% 0.30/0.46 => ( number1551313001itions @ P2 @ N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitionsI
% 0.30/0.46 thf(fact_63_partitionsE,axiom,
% 0.30/0.46 ! [P2: nat > nat,N2: nat] :
% 0.30/0.46 ( ( number1551313001itions @ P2 @ N2 )
% 0.30/0.46 => ~ ( ! [I4: nat] :
% 0.30/0.46 ( ( ( P2 @ I4 )
% 0.30/0.46 != zero_zero_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ one_one_nat @ I4 )
% 0.30/0.46 & ( ord_less_eq_nat @ I4 @ N2 ) ) )
% 0.30/0.46 => ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [I: nat] : ( times_times_nat @ ( P2 @ I ) @ I )
% 0.30/0.46 @ ( set_ord_atMost_nat @ N2 ) )
% 0.30/0.46 != N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitionsE
% 0.30/0.46 thf(fact_64_le__zero__eq,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.30/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_zero_eq
% 0.30/0.46 thf(fact_65_mult__zero__left,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % mult_zero_left
% 0.30/0.46 thf(fact_66_mult__zero__right,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % mult_zero_right
% 0.30/0.46 thf(fact_67_mult__eq__0__iff,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ A2 @ B2 )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 = ( ( A2 = zero_zero_nat )
% 0.30/0.46 | ( B2 = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_eq_0_iff
% 0.30/0.46 thf(fact_68_mult__cancel__left,axiom,
% 0.30/0.46 ! [C: nat,A2: nat,B2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ C @ A2 )
% 0.30/0.46 = ( times_times_nat @ C @ B2 ) )
% 0.30/0.46 = ( ( C = zero_zero_nat )
% 0.30/0.46 | ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_left
% 0.30/0.46 thf(fact_69_mult__cancel__right,axiom,
% 0.30/0.46 ! [A2: nat,C: nat,B2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ A2 @ C )
% 0.30/0.46 = ( times_times_nat @ B2 @ C ) )
% 0.30/0.46 = ( ( C = zero_zero_nat )
% 0.30/0.46 | ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel_right
% 0.30/0.46 thf(fact_70_zero__diff,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % zero_diff
% 0.30/0.46 thf(fact_71_diff__zero,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 0.30/0.46 = A2 ) ).
% 0.30/0.46
% 0.30/0.46 % diff_zero
% 0.30/0.46 thf(fact_72_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ A2 @ A2 )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.46 thf(fact_73_bot__nat__0_Oextremum,axiom,
% 0.30/0.46 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % bot_nat_0.extremum
% 0.30/0.46 thf(fact_74_le0,axiom,
% 0.30/0.46 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.30/0.46
% 0.30/0.46 % le0
% 0.30/0.46 thf(fact_75_diff__0__eq__0,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ zero_zero_nat @ N2 )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % diff_0_eq_0
% 0.30/0.46 thf(fact_76_diff__self__eq__0,axiom,
% 0.30/0.46 ! [M: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ M @ M )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % diff_self_eq_0
% 0.30/0.46 thf(fact_77_mult__is__0,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ M @ N2 )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 = ( ( M = zero_zero_nat )
% 0.30/0.46 | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_is_0
% 0.30/0.46 thf(fact_78_mult__0__right,axiom,
% 0.30/0.46 ! [M: nat] :
% 0.30/0.46 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % mult_0_right
% 0.30/0.46 thf(fact_79_mult__cancel1,axiom,
% 0.30/0.46 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ K @ M )
% 0.30/0.46 = ( times_times_nat @ K @ N2 ) )
% 0.30/0.46 = ( ( M = N2 )
% 0.30/0.46 | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel1
% 0.30/0.46 thf(fact_80_mult__cancel2,axiom,
% 0.30/0.46 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ M @ K )
% 0.30/0.46 = ( times_times_nat @ N2 @ K ) )
% 0.30/0.46 = ( ( M = N2 )
% 0.30/0.46 | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_cancel2
% 0.30/0.46 thf(fact_81_sum_Oneutral__const,axiom,
% 0.30/0.46 ! [A: set_nat] :
% 0.30/0.46 ( ( groups1842438620at_nat
% 0.30/0.46 @ ^ [Uu: nat] : zero_zero_nat
% 0.30/0.46 @ A )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % sum.neutral_const
% 0.30/0.46 thf(fact_82_diff__is__0__eq_H,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.46 => ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.46 = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_is_0_eq'
% 0.30/0.46 thf(fact_83_diff__is__0__eq,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_is_0_eq
% 0.30/0.46 thf(fact_84_gr0,axiom,
% 0.30/0.46 ord_less_nat @ zero_zero_nat @ ( p @ k ) ).
% 0.30/0.46
% 0.30/0.46 % gr0
% 0.30/0.46 thf(fact_85_zero__reorient,axiom,
% 0.30/0.46 ! [X: nat] :
% 0.30/0.46 ( ( zero_zero_nat = X )
% 0.30/0.46 = ( X = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_reorient
% 0.30/0.46 thf(fact_86_zero__le,axiom,
% 0.30/0.46 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le
% 0.30/0.46 thf(fact_87_mult__not__zero,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ A2 @ B2 )
% 0.30/0.46 != zero_zero_nat )
% 0.30/0.46 => ( ( A2 != zero_zero_nat )
% 0.30/0.46 & ( B2 != zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_not_zero
% 0.30/0.46 thf(fact_88_divisors__zero,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ( times_times_nat @ A2 @ B2 )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 => ( ( A2 = zero_zero_nat )
% 0.30/0.46 | ( B2 = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % divisors_zero
% 0.30/0.46 thf(fact_89_no__zero__divisors,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( A2 != zero_zero_nat )
% 0.30/0.46 => ( ( B2 != zero_zero_nat )
% 0.30/0.46 => ( ( times_times_nat @ A2 @ B2 )
% 0.30/0.46 != zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % no_zero_divisors
% 0.30/0.46 thf(fact_90_mult__left__cancel,axiom,
% 0.30/0.46 ! [C: nat,A2: nat,B2: nat] :
% 0.30/0.46 ( ( C != zero_zero_nat )
% 0.30/0.46 => ( ( ( times_times_nat @ C @ A2 )
% 0.30/0.46 = ( times_times_nat @ C @ B2 ) )
% 0.30/0.46 = ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_left_cancel
% 0.30/0.46 thf(fact_91_mult__right__cancel,axiom,
% 0.30/0.46 ! [C: nat,A2: nat,B2: nat] :
% 0.30/0.46 ( ( C != zero_zero_nat )
% 0.30/0.46 => ( ( ( times_times_nat @ A2 @ C )
% 0.30/0.46 = ( times_times_nat @ B2 @ C ) )
% 0.30/0.46 = ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_right_cancel
% 0.30/0.46 thf(fact_92_zero__neq__one,axiom,
% 0.30/0.46 zero_zero_nat != one_one_nat ).
% 0.30/0.46
% 0.30/0.46 % zero_neq_one
% 0.30/0.46 thf(fact_93_lambda__zero,axiom,
% 0.30/0.46 ( ( ^ [H2: nat] : zero_zero_nat )
% 0.30/0.46 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % lambda_zero
% 0.30/0.46 thf(fact_94_sum_Oneutral,axiom,
% 0.30/0.46 ! [A: set_nat,G: nat > nat] :
% 0.30/0.46 ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ A )
% 0.30/0.46 => ( ( G @ X2 )
% 0.30/0.46 = zero_zero_nat ) )
% 0.30/0.46 => ( ( groups1842438620at_nat @ G @ A )
% 0.30/0.46 = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.neutral
% 0.30/0.46 thf(fact_95_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.46 ! [G: nat > nat,A: set_nat] :
% 0.30/0.46 ( ( ( groups1842438620at_nat @ G @ A )
% 0.30/0.46 != zero_zero_nat )
% 0.30/0.46 => ~ ! [A3: nat] :
% 0.30/0.46 ( ( member_nat @ A3 @ A )
% 0.30/0.46 => ( ( G @ A3 )
% 0.30/0.46 = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.not_neutral_contains_not_neutral
% 0.30/0.46 thf(fact_96_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.30/0.46 => ( A2 = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_nat_0.extremum_uniqueI
% 0.30/0.46 thf(fact_97_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.46 ! [A2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.30/0.46 = ( A2 = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % bot_nat_0.extremum_unique
% 0.30/0.46 thf(fact_98_le__0__eq,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.30/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_0_eq
% 0.30/0.46 thf(fact_99_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.46 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.30/0.46
% 0.30/0.46 % less_eq_nat.simps(1)
% 0.30/0.46 thf(fact_100_diffs0__imp__equal,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 => ( ( ( minus_minus_nat @ N2 @ M )
% 0.30/0.46 = zero_zero_nat )
% 0.30/0.46 => ( M = N2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diffs0_imp_equal
% 0.30/0.46 thf(fact_101_minus__nat_Odiff__0,axiom,
% 0.30/0.46 ! [M: nat] :
% 0.30/0.46 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.30/0.46 = M ) ).
% 0.30/0.46
% 0.30/0.46 % minus_nat.diff_0
% 0.30/0.46 thf(fact_102_mult__0,axiom,
% 0.30/0.46 ! [N2: nat] :
% 0.30/0.46 ( ( times_times_nat @ zero_zero_nat @ N2 )
% 0.30/0.46 = zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % mult_0
% 0.30/0.46 thf(fact_103_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.46 thf(fact_104_mult__nonneg__nonpos2,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonneg_nonpos2
% 0.30/0.46 thf(fact_105_mult__nonpos__nonneg,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonpos_nonneg
% 0.30/0.46 thf(fact_106_mult__nonneg__nonpos,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonneg_nonpos
% 0.30/0.46 thf(fact_107_mult__nonneg__nonneg,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.46 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_nonneg_nonneg
% 0.30/0.46 thf(fact_108_split__mult__neg__le,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
% 0.30/0.46 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.30/0.46 & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_mult_neg_le
% 0.30/0.46 thf(fact_109_mult__right__mono,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_right_mono
% 0.30/0.46 thf(fact_110_mult__left__mono,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_left_mono
% 0.30/0.46 thf(fact_111_mult__mono_H,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_mono'
% 0.30/0.46 thf(fact_112_mult__mono,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_mono
% 0.30/0.46 thf(fact_113_not__one__le__zero,axiom,
% 0.30/0.46 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.46
% 0.30/0.46 % not_one_le_zero
% 0.30/0.46 thf(fact_114_zero__le__one,axiom,
% 0.30/0.46 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.46
% 0.30/0.46 % zero_le_one
% 0.30/0.46 thf(fact_115_partitions__zero,axiom,
% 0.30/0.46 ! [P2: nat > nat] :
% 0.30/0.46 ( ( number1551313001itions @ P2 @ zero_zero_nat )
% 0.30/0.46 = ( P2
% 0.30/0.46 = ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % partitions_zero
% 0.30/0.46 thf(fact_116_mult__left__le,axiom,
% 0.30/0.46 ! [C: nat,A2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_left_le
% 0.30/0.46 thf(fact_117_mult__le__one,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.30/0.46 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
% 0.30/0.46 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_le_one
% 0.30/0.46 thf(fact_118_sum__nonpos,axiom,
% 0.30/0.46 ! [A: set_nat,F: nat > nat] :
% 0.30/0.46 ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ A )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
% 0.30/0.46 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_nonpos
% 0.30/0.46 thf(fact_119_sum__nonneg,axiom,
% 0.30/0.46 ! [A: set_nat,F: nat > nat] :
% 0.30/0.46 ( ! [X2: nat] :
% 0.30/0.46 ( ( member_nat @ X2 @ A )
% 0.30/0.46 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
% 0.30/0.46 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1842438620at_nat @ F @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_nonneg
% 0.30/0.46 thf(fact_120_mult__eq__self__implies__10,axiom,
% 0.30/0.46 ! [M: nat,N2: nat] :
% 0.30/0.46 ( ( M
% 0.30/0.46 = ( times_times_nat @ M @ N2 ) )
% 0.30/0.46 => ( ( N2 = one_one_nat )
% 0.30/0.46 | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mult_eq_self_implies_10
% 0.30/0.46 thf(fact_121_dual__order_Oantisym,axiom,
% 0.30/0.46 ! [B2: set_nat,A2: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.antisym
% 0.30/0.46 thf(fact_122_dual__order_Oantisym,axiom,
% 0.30/0.46 ! [B2: nat,A2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.antisym
% 0.30/0.46 thf(fact_123_dual__order_Oeq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 0.30/0.46 & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.eq_iff
% 0.30/0.46 thf(fact_124_dual__order_Oeq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.30/0.46 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.eq_iff
% 0.30/0.46 thf(fact_125_dual__order_Otrans,axiom,
% 0.30/0.46 ! [B2: set_nat,A2: set_nat,C: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ C @ B2 )
% 0.30/0.46 => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.trans
% 0.30/0.46 thf(fact_126_dual__order_Otrans,axiom,
% 0.30/0.46 ! [B2: nat,A2: nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ C @ B2 )
% 0.30/0.46 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.trans
% 0.30/0.46 thf(fact_127_linorder__wlog,axiom,
% 0.30/0.46 ! [P: nat > nat > $o,A2: nat,B2: nat] :
% 0.30/0.46 ( ! [A3: nat,B3: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.30/0.46 => ( P @ A3 @ B3 ) )
% 0.30/0.46 => ( ! [A3: nat,B3: nat] :
% 0.30/0.46 ( ( P @ B3 @ A3 )
% 0.30/0.46 => ( P @ A3 @ B3 ) )
% 0.30/0.46 => ( P @ A2 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % linorder_wlog
% 0.30/0.46 thf(fact_128_dual__order_Orefl,axiom,
% 0.30/0.46 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.refl
% 0.30/0.46 thf(fact_129_dual__order_Orefl,axiom,
% 0.30/0.46 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.30/0.46
% 0.30/0.46 % dual_order.refl
% 0.30/0.46 thf(fact_130_order__trans,axiom,
% 0.30/0.46 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 0.30/0.46 => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_trans
% 0.30/0.46 thf(fact_131_order__trans,axiom,
% 0.30/0.46 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.30/0.46 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_trans
% 0.30/0.46 thf(fact_132_order__class_Oorder_Oantisym,axiom,
% 0.30/0.46 ! [A2: set_nat,B2: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.antisym
% 0.30/0.46 thf(fact_133_order__class_Oorder_Oantisym,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.30/0.46 => ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.antisym
% 0.30/0.46 thf(fact_134_ord__le__eq__trans,axiom,
% 0.30/0.46 ! [A2: set_nat,B2: set_nat,C: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( B2 = C )
% 0.30/0.46 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_trans
% 0.30/0.46 thf(fact_135_ord__le__eq__trans,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( B2 = C )
% 0.30/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_trans
% 0.30/0.46 thf(fact_136_ord__eq__le__trans,axiom,
% 0.30/0.46 ! [A2: set_nat,B2: set_nat,C: set_nat] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.30/0.46 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_trans
% 0.30/0.46 thf(fact_137_ord__eq__le__trans,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_trans
% 0.30/0.46 thf(fact_138_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.30/0.46 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.eq_iff
% 0.30/0.46 thf(fact_139_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [A4: nat,B4: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.30/0.46 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.eq_iff
% 0.30/0.46 thf(fact_140_antisym__conv,axiom,
% 0.30/0.46 ! [Y: set_nat,X: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ X @ Y )
% 0.30/0.46 = ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_141_antisym__conv,axiom,
% 0.30/0.46 ! [Y: nat,X: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ Y @ X )
% 0.30/0.46 => ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.46 = ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_142_le__cases3,axiom,
% 0.30/0.46 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.46 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.46 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.30/0.46 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.30/0.46 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.30/0.46 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.30/0.46 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.30/0.46 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.30/0.46 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.30/0.46 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.30/0.46 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.30/0.46 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.30/0.46 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases3
% 0.30/0.46 thf(fact_143_order_Otrans,axiom,
% 0.30/0.46 ! [A2: set_nat,B2: set_nat,C: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.30/0.46 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_144_order_Otrans,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.46 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_145_le__cases,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.30/0.46 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases
% 0.30/0.46 thf(fact_146_eq__refl,axiom,
% 0.30/0.46 ! [X: set_nat,Y: set_nat] :
% 0.30/0.46 ( ( X = Y )
% 0.30/0.46 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_147_eq__refl,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ( X = Y )
% 0.30/0.46 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_148_linear,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.46 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % linear
% 0.30/0.46 thf(fact_149_antisym,axiom,
% 0.30/0.46 ! [X: set_nat,Y: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ Y @ X )
% 0.30/0.46 => ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_150_antisym,axiom,
% 0.30/0.46 ! [X: nat,Y: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_nat @ Y @ X )
% 0.30/0.46 => ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_151_eq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [X4: set_nat,Y4: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ X4 @ Y4 )
% 0.30/0.46 & ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_152_eq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 0.30/0.46 = ( ^ [X4: nat,Y4: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.30/0.46 & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_153_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ( F @ B2 )
% 0.30/0.46 = C )
% 0.30/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_154_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ( F @ B2 )
% 0.30/0.46 = C )
% 0.30/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_155_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ( F @ B2 )
% 0.30/0.46 = C )
% 0.30/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_156_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.46 => ( ( ( F @ B2 )
% 0.30/0.46 = C )
% 0.30/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_157_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
% 0.30/0.46 ( ( A2
% 0.30/0.46 = ( F @ B2 ) )
% 0.30/0.46 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_158_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
% 0.30/0.46 ( ( A2
% 0.30/0.46 = ( F @ B2 ) )
% 0.30/0.46 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.30/0.46 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.46 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.46 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_159_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_subst
% 0.30/0.47 thf(fact_160_ord__eq__le__subst,axiom,
% 0.30/0.47 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_subst
% 0.30/0.47 thf(fact_161_order__subst2,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_162_order__subst2,axiom,
% 0.30/0.47 ! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 0.30/0.47 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_163_order__subst2,axiom,
% 0.30/0.47 ! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
% 0.30/0.47 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_164_order__subst2,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_165_order__subst1,axiom,
% 0.30/0.47 ! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_166_order__subst1,axiom,
% 0.30/0.47 ! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_167_order__subst1,axiom,
% 0.30/0.47 ! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: set_nat,Y2: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_168_order__subst1,axiom,
% 0.30/0.47 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_169_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
% 0.30/0.47 = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.47 thf(fact_170_mult_Oassoc,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
% 0.30/0.47 = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.assoc
% 0.30/0.47 thf(fact_171_mult_Ocommute,axiom,
% 0.30/0.47 ( times_times_nat
% 0.30/0.47 = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.commute
% 0.30/0.47 thf(fact_172_mult_Oleft__commute,axiom,
% 0.30/0.47 ! [B2: nat,A2: nat,C: nat] :
% 0.30/0.47 ( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
% 0.30/0.47 = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult.left_commute
% 0.30/0.47 thf(fact_173_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.30/0.47 ! [A2: nat,C: nat,B2: nat] :
% 0.30/0.47 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
% 0.30/0.47 = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% 0.30/0.47
% 0.30/0.47 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.30/0.47 thf(fact_174_one__reorient,axiom,
% 0.30/0.47 ! [X: nat] :
% 0.30/0.47 ( ( one_one_nat = X )
% 0.30/0.47 = ( X = one_one_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % one_reorient
% 0.30/0.47 thf(fact_175_fun__upd__idem__iff,axiom,
% 0.30/0.47 ! [F: nat > nat,X: nat,Y: nat] :
% 0.30/0.47 ( ( ( fun_upd_nat_nat @ F @ X @ Y )
% 0.30/0.47 = F )
% 0.30/0.47 = ( ( F @ X )
% 0.30/0.47 = Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_idem_iff
% 0.30/0.47 thf(fact_176_fun__upd__twist,axiom,
% 0.30/0.47 ! [A2: nat,C: nat,M: nat > nat,B2: nat,D: nat] :
% 0.30/0.47 ( ( A2 != C )
% 0.30/0.47 => ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ A2 @ B2 ) @ C @ D )
% 0.30/0.47 = ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_twist
% 0.30/0.47 thf(fact_177_fun__upd__other,axiom,
% 0.30/0.47 ! [Z: nat,X: nat,F: nat > nat,Y: nat] :
% 0.30/0.47 ( ( Z != X )
% 0.30/0.47 => ( ( fun_upd_nat_nat @ F @ X @ Y @ Z )
% 0.30/0.47 = ( F @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_other
% 0.30/0.47 thf(fact_178_fun__upd__same,axiom,
% 0.30/0.47 ! [F: nat > nat,X: nat,Y: nat] :
% 0.30/0.47 ( ( fun_upd_nat_nat @ F @ X @ Y @ X )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_same
% 0.30/0.47 thf(fact_179_fun__upd__idem,axiom,
% 0.30/0.47 ! [F: nat > nat,X: nat,Y: nat] :
% 0.30/0.47 ( ( ( F @ X )
% 0.30/0.47 = Y )
% 0.30/0.47 => ( ( fun_upd_nat_nat @ F @ X @ Y )
% 0.30/0.47 = F ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_idem
% 0.30/0.47 thf(fact_180_fun__upd__eqD,axiom,
% 0.30/0.47 ! [F: nat > nat,X: nat,Y: nat,G: nat > nat,Z: nat] :
% 0.30/0.47 ( ( ( fun_upd_nat_nat @ F @ X @ Y )
% 0.30/0.47 = ( fun_upd_nat_nat @ G @ X @ Z ) )
% 0.30/0.47 => ( Y = Z ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_eqD
% 0.30/0.47 thf(fact_181_fun__upd__def,axiom,
% 0.30/0.47 ( fun_upd_nat_nat
% 0.30/0.47 = ( ^ [F2: nat > nat,A4: nat,B4: nat,X4: nat] : ( if_nat @ ( X4 = A4 ) @ B4 @ ( F2 @ X4 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % fun_upd_def
% 0.30/0.47 thf(fact_182_left__diff__distrib_H,axiom,
% 0.30/0.47 ! [B2: nat,C: nat,A2: nat] :
% 0.30/0.47 ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A2 )
% 0.30/0.47 = ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % left_diff_distrib'
% 0.30/0.47 thf(fact_183_right__diff__distrib_H,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C ) )
% 0.30/0.47 = ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % right_diff_distrib'
% 0.30/0.47 thf(fact_184_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.47 ! [A2: nat] :
% 0.30/0.47 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.30/0.47 = A2 ) ).
% 0.30/0.47
% 0.30/0.47 % comm_monoid_mult_class.mult_1
% 0.30/0.47 thf(fact_185_mult_Ocomm__neutral,axiom,
% 0.30/0.47 ! [A2: nat] :
% 0.30/0.47 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.30/0.47 = A2 ) ).
% 0.30/0.47
% 0.30/0.47 % mult.comm_neutral
% 0.30/0.47 thf(fact_186_subsetI,axiom,
% 0.30/0.47 ! [A: set_nat,B: set_nat] :
% 0.30/0.47 ( ! [X2: nat] :
% 0.30/0.47 ( ( member_nat @ X2 @ A )
% 0.30/0.47 => ( member_nat @ X2 @ B ) )
% 0.30/0.47 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 0.30/0.47
% 0.30/0.47 % subsetI
% 0.30/0.47 thf(fact_187_subset__antisym,axiom,
% 0.30/0.47 ! [A: set_nat,B: set_nat] :
% 0.30/0.47 ( ( ord_less_eq_set_nat @ A @ B )
% 0.30/0.47 => ( ( ord_less_eq_set_nat @ B @ A )
% 0.30/0.47 => ( A = B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % subset_antisym
% 0.30/0.47 thf(fact_188_partitions__remove1__bounds,axiom,
% 0.30/0.47 ! [P2: nat > nat,N2: nat,K: nat,I2: nat] :
% 0.30/0.47 ( ( number1551313001itions @ P2 @ N2 )
% 0.30/0.47 => ( ( ord_less_nat @ zero_zero_nat @ ( P2 @ K ) )
% 0.30/0.47 => ( ( ( fun_upd_nat_nat @ P2 @ K @ ( minus_minus_nat @ ( P2 @ K ) @ one_one_nat ) @ I2 )
% 0.30/0.47 != zero_zero_nat )
% 0.30/0.47 => ( ( ord_less_eq_nat @ one_one_nat @ I2 )
% 0.30/0.47 & ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % partitions_remove1_bounds
% 0.30/0.47 thf(fact_189_nat__mult__eq__cancel__disj,axiom,
% 0.30/0.47 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.47 ( ( ( times_times_nat @ K @ M )
% 0.30/0.47 = ( times_times_nat @ K @ N2 ) )
% 0.30/0.47 = ( ( K = zero_zero_nat )
% 0.30/0.47 | ( M = N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_mult_eq_cancel_disj
% 0.30/0.47 thf(fact_190_le__numeral__extra_I4_J,axiom,
% 0.30/0.47 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(4)
% 0.30/0.47 thf(fact_191_le__numeral__extra_I3_J,axiom,
% 0.30/0.47 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.30/0.47
% 0.30/0.47 % le_numeral_extra(3)
% 0.30/0.47 thf(fact_192_not__gr__zero,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.47 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_gr_zero
% 0.30/0.47 thf(fact_193_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.47 ! [A2: nat] :
% 0.30/0.47 ( ( A2 != zero_zero_nat )
% 0.30/0.47 = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % bot_nat_0.not_eq_extremum
% 0.30/0.47 thf(fact_194_less__nat__zero__code,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % less_nat_zero_code
% 0.30/0.47 thf(fact_195_neq0__conv,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ( ( N2 != zero_zero_nat )
% 0.30/0.47 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq0_conv
% 0.30/0.47 thf(fact_196_zero__less__diff,axiom,
% 0.30/0.47 ! [N2: nat,M: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
% 0.30/0.47 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % zero_less_diff
% 0.30/0.47 thf(fact_197_nat__mult__less__cancel__disj,axiom,
% 0.30/0.47 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.47 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_mult_less_cancel_disj
% 0.30/0.47 thf(fact_198_mult__less__cancel2,axiom,
% 0.30/0.47 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 0.30/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.47 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_less_cancel2
% 0.30/0.47 thf(fact_199_nat__0__less__mult__iff,axiom,
% 0.30/0.47 ! [M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.47 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.47 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_0_less_mult_iff
% 0.30/0.47 thf(fact_200_less__one,axiom,
% 0.30/0.47 ! [N2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ N2 @ one_one_nat )
% 0.30/0.47 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_one
% 0.30/0.47 thf(fact_201_nat__mult__le__cancel__disj,axiom,
% 0.30/0.47 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_mult_le_cancel_disj
% 0.30/0.47 thf(fact_202_mult__le__cancel2,axiom,
% 0.30/0.47 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 0.30/0.47 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.47 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % mult_le_cancel2
% 0.30/0.47 thf(fact_203_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [B2: nat,A2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ B2 @ A2 )
% 0.30/0.47 => ( A2 != B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_not_eq
% 0.30/0.47 thf(fact_204_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ( A2 != B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_not_eq
% 0.30/0.47 thf(fact_205_not__less__iff__gr__or__eq,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.30/0.47 = ( ( ord_less_nat @ Y @ X )
% 0.30/0.47 | ( X = Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less_iff_gr_or_eq
% 0.30/0.47 thf(fact_206_dual__order_Ostrict__trans,axiom,
% 0.30/0.47 ! [B2: nat,A2: nat,C: nat] :
% 0.30/0.47 ( ( ord_less_nat @ B2 @ A2 )
% 0.30/0.47 => ( ( ord_less_nat @ C @ B2 )
% 0.30/0.47 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans
% 0.30/0.47 thf(fact_207_linorder__less__wlog,axiom,
% 0.30/0.47 ! [P: nat > nat > $o,A2: nat,B2: nat] :
% 0.30/0.47 ( ! [A3: nat,B3: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A3 @ B3 )
% 0.30/0.47 => ( P @ A3 @ B3 ) )
% 0.30/0.47 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 0.30/0.47 => ( ! [A3: nat,B3: nat] :
% 0.30/0.47 ( ( P @ B3 @ A3 )
% 0.30/0.47 => ( P @ A3 @ B3 ) )
% 0.30/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_less_wlog
% 0.30/0.47 thf(fact_208_exists__least__iff,axiom,
% 0.30/0.47 ( ( ^ [P4: nat > $o] :
% 0.30/0.47 ? [X5: nat] : ( P4 @ X5 ) )
% 0.30/0.47 = ( ^ [P5: nat > $o] :
% 0.30/0.47 ? [N: nat] :
% 0.30/0.47 ( ( P5 @ N )
% 0.30/0.47 & ! [M4: nat] :
% 0.30/0.47 ( ( ord_less_nat @ M4 @ N )
% 0.30/0.47 => ~ ( P5 @ M4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % exists_least_iff
% 0.30/0.47 thf(fact_209_less__imp__not__less,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_less
% 0.30/0.47 thf(fact_210_order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.47 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans
% 0.30/0.47 thf(fact_211_dual__order_Oirrefl,axiom,
% 0.30/0.47 ! [A2: nat] :
% 0.30/0.47 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.irrefl
% 0.30/0.47 thf(fact_212_linorder__cases,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ~ ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( ( X != Y )
% 0.30/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_cases
% 0.30/0.47 thf(fact_213_less__imp__triv,axiom,
% 0.30/0.47 ! [X: nat,Y: nat,P: $o] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( ( ord_less_nat @ Y @ X )
% 0.30/0.47 => P ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_triv
% 0.30/0.47 thf(fact_214_less__imp__not__eq2,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( Y != X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq2
% 0.30/0.47 thf(fact_215_antisym__conv3,axiom,
% 0.30/0.47 ! [Y: nat,X: nat] :
% 0.30/0.47 ( ~ ( ord_less_nat @ Y @ X )
% 0.30/0.47 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.30/0.47 = ( X = Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv3
% 0.30/0.47 thf(fact_216_less__induct,axiom,
% 0.30/0.47 ! [P: nat > $o,A2: nat] :
% 0.30/0.47 ( ! [X2: nat] :
% 0.30/0.47 ( ! [Y3: nat] :
% 0.30/0.47 ( ( ord_less_nat @ Y3 @ X2 )
% 0.30/0.47 => ( P @ Y3 ) )
% 0.30/0.47 => ( P @ X2 ) )
% 0.30/0.47 => ( P @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_induct
% 0.30/0.47 thf(fact_217_less__not__sym,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_not_sym
% 0.30/0.47 thf(fact_218_less__imp__not__eq,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( X != Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq
% 0.30/0.47 thf(fact_219_dual__order_Oasym,axiom,
% 0.30/0.47 ! [B2: nat,A2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ B2 @ A2 )
% 0.30/0.47 => ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.asym
% 0.30/0.47 thf(fact_220_ord__less__eq__trans,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( B2 = C )
% 0.30/0.47 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_trans
% 0.30/0.47 thf(fact_221_ord__eq__less__trans,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( A2 = B2 )
% 0.30/0.47 => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.47 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_trans
% 0.30/0.47 thf(fact_222_less__irrefl,axiom,
% 0.30/0.47 ! [X: nat] :
% 0.30/0.47 ~ ( ord_less_nat @ X @ X ) ).
% 0.30/0.47
% 0.30/0.47 % less_irrefl
% 0.30/0.47 thf(fact_223_less__linear,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 | ( X = Y )
% 0.30/0.47 | ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_linear
% 0.30/0.47 thf(fact_224_less__trans,axiom,
% 0.30/0.47 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( ( ord_less_nat @ Y @ Z )
% 0.30/0.47 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_trans
% 0.30/0.47 thf(fact_225_less__asym_H,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym'
% 0.30/0.47 thf(fact_226_less__asym,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_asym
% 0.30/0.47 thf(fact_227_less__imp__neq,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( X != Y ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_neq
% 0.30/0.47 thf(fact_228_order_Oasym,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.asym
% 0.30/0.47 thf(fact_229_neq__iff,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 = ( ( ord_less_nat @ X @ Y )
% 0.30/0.47 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neq_iff
% 0.30/0.47 thf(fact_230_neqE,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % neqE
% 0.30/0.47 thf(fact_231_gt__ex,axiom,
% 0.30/0.47 ! [X: nat] :
% 0.30/0.47 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.30/0.47
% 0.30/0.47 % gt_ex
% 0.30/0.47 thf(fact_232_order__less__subst2,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst2
% 0.30/0.47 thf(fact_233_order__less__subst1,axiom,
% 0.30/0.47 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_less_subst1
% 0.30/0.47 thf(fact_234_ord__less__eq__subst,axiom,
% 0.30/0.47 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 0.30/0.47 ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.47 => ( ( ( F @ B2 )
% 0.30/0.47 = C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_less_eq_subst
% 0.30/0.47 thf(fact_235_ord__eq__less__subst,axiom,
% 0.30/0.47 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.47 => ( ! [X2: nat,Y2: nat] :
% 0.30/0.47 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/0.47 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/0.47 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_less_subst
% 0.30/0.47 thf(fact_236_less__numeral__extra_I3_J,axiom,
% 0.30/0.47 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.30/0.47
% 0.30/0.47 % less_numeral_extra(3)
% 0.30/0.47 thf(fact_237_less__numeral__extra_I4_J,axiom,
% 0.30/0.47 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/0.47
% 0.30/0.47 % less_numeral_extra(4)
% 0.30/0.47 thf(fact_238_linorder__neqE__nat,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( X != Y )
% 0.30/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.30/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_neqE_nat
% 0.30/0.47 thf(fact_239_infinite__descent,axiom,
% 0.30/0.47 ! [P: nat > $o,N2: nat] :
% 0.30/0.47 ( ! [N3: nat] :
% 0.30/0.47 ( ~ ( P @ N3 )
% 0.30/0.47 => ? [M5: nat] :
% 0.30/0.47 ( ( ord_less_nat @ M5 @ N3 )
% 0.30/0.47 & ~ ( P @ M5 ) ) )
% 0.30/0.47 => ( P @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % infinite_descent
% 0.30/0.47 thf(fact_240_nat__less__induct,axiom,
% 0.30/0.47 ! [P: nat > $o,N2: nat] :
% 0.30/0.47 ( ! [N3: nat] :
% 0.30/0.47 ( ! [M5: nat] :
% 0.30/0.47 ( ( ord_less_nat @ M5 @ N3 )
% 0.30/0.47 => ( P @ M5 ) )
% 0.30/0.47 => ( P @ N3 ) )
% 0.30/0.47 => ( P @ N2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % nat_less_induct
% 0.30/0.47
% 0.30/0.47 % Helper facts (3)
% 0.30/0.47 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.47 ! [P: $o] :
% 0.30/0.47 ( ( P = $true )
% 0.30/0.47 | ( P = $false ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( if_nat @ $false @ X @ Y )
% 0.30/0.47 = Y ) ).
% 0.30/0.47
% 0.30/0.47 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.47 ! [X: nat,Y: nat] :
% 0.30/0.47 ( ( if_nat @ $true @ X @ Y )
% 0.30/0.47 = X ) ).
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 ( ( minus_minus_nat
% 0.30/0.47 @ ( groups1842438620at_nat
% 0.30/0.47 @ ^ [I: nat] : ( times_times_nat @ ( p @ I ) @ I )
% 0.30/0.47 @ ( set_ord_atMost_nat @ n ) )
% 0.30/0.47 @ k )
% 0.30/0.47 = ( minus_minus_nat @ n @ k ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.52/1.53 ---- Embedded in THF ---
% 2.52/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.52/1.53 %%% Generated on Tue Feb 24 08:27:18 EST 2026
% 2.52/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.52/1.53 %%% Logic specification used:
% 2.52/1.53 %%% thf(spec, logic, $$dhol).
% 2.52/1.53
% 2.52/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2DTF2249.p
% See solution above
% 2.52/1.53 ------------------------
% 2.52/1.55 ---- Cleaned THF ---
% 2.52/1.55 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.52/1.55 set_set_nat: $tType ).
% 2.52/1.55
% 2.52/1.55 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.52/1.55 set_nat: $tType ).
% 2.52/1.55
% 2.52/1.55 thf(ty_n_t__Nat__Onat,type,
% 2.52/1.55 nat: $tType ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.52/1.55 fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 2.52/1.55 minus_minus_nat: nat > nat > nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.52/1.55 one_one_nat: nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 2.52/1.55 times_times_nat: nat > nat > nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.52/1.55 zero_zero_nat: nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.52/1.55 groups1842438620at_nat: ( nat > nat ) > set_nat > nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_If_001t__Nat__Onat,type,
% 2.52/1.55 if_nat: $o > nat > nat > nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Number__Partition__Mirabelle__zerdlymyoj_Opartitions,type,
% 2.52/1.55 number1551313001itions: ( nat > nat ) > nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.52/1.55 ord_less_nat: nat > nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.52/1.55 ord_less_eq_nat: nat > nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.52/1.55 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.52/1.55 ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 2.52/1.55 collect_nat: ( nat > $o ) > set_nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.52/1.55 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 2.52/1.55 set_ord_atMost_nat: nat > set_nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.52/1.55 set_or1086813439et_nat: set_nat > set_set_nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_member_001t__Nat__Onat,type,
% 2.52/1.55 member_nat: nat > set_nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.52/1.55 member_set_nat: set_nat > set_set_nat > $o ).
% 2.52/1.55
% 2.52/1.55 thf(sy_v_k,type,
% 2.52/1.55 k: nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_v_n,type,
% 2.52/1.55 n: nat ).
% 2.52/1.55
% 2.52/1.55 thf(sy_v_p,type,
% 2.52/1.55 p: nat > nat ).
% 2.52/1.55
% 2.52/1.55 thf(fact_0_partitions,axiom,
% 2.52/1.55 number1551313001itions @ p @ n ).
% 2.52/1.55
% 2.52/1.55 thf(fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062,axiom,
% 2.52/1.55 ord_less_eq_nat @ k @ n ).
% 2.52/1.55
% 2.52/1.55 thf(fact_2_atMost__eq__iff,axiom,
% 2.52/1.55 ! [X: nat,Y: nat] :
% 2.52/1.55 ( ( ( set_ord_atMost_nat @ X )
% 2.52/1.55 = ( set_ord_atMost_nat @ Y ) )
% 2.52/1.55 = ( X = Y ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_3__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062,axiom,
% 2.52/1.55 ( ( groups1842438620at_nat
% 2.52/1.55 @ ^ [I: nat] : ( times_times_nat @ ( fun_upd_nat_nat @ p @ k @ ( minus_minus_nat @ ( p @ k ) @ one_one_nat ) @ I ) @ I )
% 2.52/1.55 @ ( set_ord_atMost_nat @ ( minus_minus_nat @ n @ k ) ) )
% 2.52/1.55 = ( groups1842438620at_nat
% 2.52/1.55 @ ^ [I: nat] : ( times_times_nat @ ( fun_upd_nat_nat @ p @ k @ ( minus_minus_nat @ ( p @ k ) @ one_one_nat ) @ I ) @ I )
% 2.52/1.55 @ ( set_ord_atMost_nat @ n ) ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_4_calculation,axiom,
% 2.52/1.55 ( ( groups1842438620at_nat
% 2.52/1.55 @ ^ [I: nat] : ( times_times_nat @ ( fun_upd_nat_nat @ p @ k @ ( minus_minus_nat @ ( p @ k ) @ one_one_nat ) @ I ) @ I )
% 2.52/1.55 @ ( set_ord_atMost_nat @ ( minus_minus_nat @ n @ k ) ) )
% 2.52/1.55 = ( minus_minus_nat
% 2.52/1.55 @ ( groups1842438620at_nat
% 2.52/1.55 @ ^ [I: nat] : ( times_times_nat @ ( p @ I ) @ I )
% 2.52/1.55 @ ( set_ord_atMost_nat @ n ) )
% 2.52/1.55 @ k ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_5_sum__product,axiom,
% 2.52/1.55 ! [F: nat > nat,A: set_nat,G: nat > nat,B: set_nat] :
% 2.52/1.55 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ B ) )
% 2.52/1.55 = ( groups1842438620at_nat
% 2.52/1.55 @ ^ [I: nat] :
% 2.52/1.55 ( groups1842438620at_nat
% 2.52/1.55 @ ^ [J: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J ) )
% 2.52/1.55 @ B )
% 2.52/1.55 @ A ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_6_sum__distrib__left,axiom,
% 2.52/1.55 ! [R: nat,F: nat > nat,A: set_nat] :
% 2.52/1.55 ( ( times_times_nat @ R @ ( groups1842438620at_nat @ F @ A ) )
% 2.52/1.55 = ( groups1842438620at_nat
% 2.52/1.55 @ ^ [N: nat] : ( times_times_nat @ R @ ( F @ N ) )
% 2.52/1.55 @ A ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_7_sum__distrib__right,axiom,
% 2.52/1.55 ! [F: nat > nat,A: set_nat,R: nat] :
% 2.52/1.55 ( ( times_times_nat @ ( groups1842438620at_nat @ F @ A ) @ R )
% 2.52/1.55 = ( groups1842438620at_nat
% 2.52/1.55 @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R )
% 2.52/1.55 @ A ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_8_diff__mult__distrib,axiom,
% 2.52/1.55 ! [M: nat,N2: nat,K: nat] :
% 2.52/1.55 ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
% 2.52/1.55 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_9_diff__mult__distrib2,axiom,
% 2.52/1.55 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.55 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 2.52/1.55 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 2.52/1.55
% 2.52/1.55 thf(fact_10_atMost__subset__iff,axiom,
% 2.52/1.55 ! [X: set_nat,Y: set_nat] :
% 2.52/1.55 ( ( ord_le1613022364et_nat @ ( set_or1086813439et_nat @ X ) @ ( set_or1086813439et_nat @ Y ) )
% 2.52/1.56 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_11_atMost__subset__iff,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
% 2.52/1.56 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_12_atMost__iff,axiom,
% 2.52/1.56 ! [I2: set_nat,K: set_nat] :
% 2.52/1.56 ( ( member_set_nat @ I2 @ ( set_or1086813439et_nat @ K ) )
% 2.52/1.56 = ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_13_atMost__iff,axiom,
% 2.52/1.56 ! [I2: nat,K: nat] :
% 2.52/1.56 ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
% 2.52/1.56 = ( ord_less_eq_nat @ I2 @ K ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_14_diff__diff__cancel,axiom,
% 2.52/1.56 ! [I2: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ I2 @ N2 )
% 2.52/1.56 => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
% 2.52/1.56 = I2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_15_nat__mult__eq__1__iff,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ M @ N2 )
% 2.52/1.56 = one_one_nat )
% 2.52/1.56 = ( ( M = one_one_nat )
% 2.52/1.56 & ( N2 = one_one_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_16_nat__1__eq__mult__iff,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( one_one_nat
% 2.52/1.56 = ( times_times_nat @ M @ N2 ) )
% 2.52/1.56 = ( ( M = one_one_nat )
% 2.52/1.56 & ( N2 = one_one_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_17_le__refl,axiom,
% 2.52/1.56 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_18_le__trans,axiom,
% 2.52/1.56 ! [I2: nat,J2: nat,K: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ I2 @ J2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ J2 @ K )
% 2.52/1.56 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_19_eq__imp__le,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( M = N2 )
% 2.52/1.56 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_20_le__antisym,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ M @ N2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 2.52/1.56 => ( M = N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_21_nat__le__linear,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ M @ N2 )
% 2.52/1.56 | ( ord_less_eq_nat @ N2 @ M ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_22_Nat_Oex__has__greatest__nat,axiom,
% 2.52/1.56 ! [P: nat > $o,K: nat,B2: nat] :
% 2.52/1.56 ( ( P @ K )
% 2.52/1.56 => ( ! [Y2: nat] :
% 2.52/1.56 ( ( P @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ Y2 @ B2 ) )
% 2.52/1.56 => ? [X2: nat] :
% 2.52/1.56 ( ( P @ X2 )
% 2.52/1.56 & ! [Y3: nat] :
% 2.52/1.56 ( ( P @ Y3 )
% 2.52/1.56 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_23_bounded__Max__nat,axiom,
% 2.52/1.56 ! [P: nat > $o,X: nat,M2: nat] :
% 2.52/1.56 ( ( P @ X )
% 2.52/1.56 => ( ! [X2: nat] :
% 2.52/1.56 ( ( P @ X2 )
% 2.52/1.56 => ( ord_less_eq_nat @ X2 @ M2 ) )
% 2.52/1.56 => ~ ! [M3: nat] :
% 2.52/1.56 ( ( P @ M3 )
% 2.52/1.56 => ~ ! [X3: nat] :
% 2.52/1.56 ( ( P @ X3 )
% 2.52/1.56 => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_24_partitions__bounds,axiom,
% 2.52/1.56 ! [P2: nat > nat,N2: nat,I2: nat] :
% 2.52/1.56 ( ( number1551313001itions @ P2 @ N2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( P2 @ I2 ) @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_25_diff__le__mono2,axiom,
% 2.52/1.56 ! [M: nat,N2: nat,L: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ M @ N2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_26_le__diff__iff_H,axiom,
% 2.52/1.56 ! [A2: nat,C: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ C )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
% 2.52/1.56 = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_27_diff__le__self,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_28_diff__le__mono,axiom,
% 2.52/1.56 ! [M: nat,N2: nat,L: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ M @ N2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_29_Nat_Odiff__diff__eq,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ K @ M )
% 2.52/1.56 => ( ( ord_less_eq_nat @ K @ N2 )
% 2.52/1.56 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 2.52/1.56 = ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_30_le__diff__iff,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ K @ M )
% 2.52/1.56 => ( ( ord_less_eq_nat @ K @ N2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 2.52/1.56 = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_31_eq__diff__iff,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ K @ M )
% 2.52/1.56 => ( ( ord_less_eq_nat @ K @ N2 )
% 2.52/1.56 => ( ( ( minus_minus_nat @ M @ K )
% 2.52/1.56 = ( minus_minus_nat @ N2 @ K ) )
% 2.52/1.56 = ( M = N2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_32_mult__le__mono2,axiom,
% 2.52/1.56 ! [I2: nat,J2: nat,K: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ I2 @ J2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_33_mult__le__mono1,axiom,
% 2.52/1.56 ! [I2: nat,J2: nat,K: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ I2 @ J2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_34_mult__le__mono,axiom,
% 2.52/1.56 ! [I2: nat,J2: nat,K: nat,L: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ I2 @ J2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ K @ L )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_35_le__square,axiom,
% 2.52/1.56 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_36_le__cube,axiom,
% 2.52/1.56 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_37_nat__mult__1__right,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( times_times_nat @ N2 @ one_one_nat )
% 2.52/1.56 = N2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_38_nat__mult__1,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( times_times_nat @ one_one_nat @ N2 )
% 2.52/1.56 = N2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_39_sum__mono,axiom,
% 2.52/1.56 ! [K2: set_nat,F: nat > nat,G: nat > nat] :
% 2.52/1.56 ( ! [I3: nat] :
% 2.52/1.56 ( ( member_nat @ I3 @ K2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ K2 ) @ ( groups1842438620at_nat @ G @ K2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_40_atMost__def,axiom,
% 2.52/1.56 ( set_or1086813439et_nat
% 2.52/1.56 = ( ^ [U: set_nat] :
% 2.52/1.56 ( collect_set_nat
% 2.52/1.56 @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_41_atMost__def,axiom,
% 2.52/1.56 ( set_ord_atMost_nat
% 2.52/1.56 = ( ^ [U: nat] :
% 2.52/1.56 ( collect_nat
% 2.52/1.56 @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_42_sum__subtractf__nat,axiom,
% 2.52/1.56 ! [A: set_nat,G: nat > nat,F: nat > nat] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
% 2.52/1.56 => ( ( groups1842438620at_nat
% 2.52/1.56 @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.52/1.56 @ A )
% 2.52/1.56 = ( minus_minus_nat @ ( groups1842438620at_nat @ F @ A ) @ ( groups1842438620at_nat @ G @ A ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_43_partitions__parts__bounded,axiom,
% 2.52/1.56 ! [P2: nat > nat,N2: nat] :
% 2.52/1.56 ( ( number1551313001itions @ P2 @ N2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ P2 @ ( set_ord_atMost_nat @ N2 ) ) @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_44_mem__Collect__eq,axiom,
% 2.52/1.56 ! [A2: nat,P: nat > $o] :
% 2.52/1.56 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 2.52/1.56 = ( P @ A2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_45_Collect__mem__eq,axiom,
% 2.52/1.56 ! [A: set_nat] :
% 2.52/1.56 ( ( collect_nat
% 2.52/1.56 @ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
% 2.52/1.56 = A ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_46_Collect__cong,axiom,
% 2.52/1.56 ! [P: nat > $o,Q: nat > $o] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ( P @ X2 )
% 2.52/1.56 = ( Q @ X2 ) )
% 2.52/1.56 => ( ( collect_nat @ P )
% 2.52/1.56 = ( collect_nat @ Q ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_47_sum_Oreindex__bij__witness,axiom,
% 2.52/1.56 ! [S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
% 2.52/1.56 ( ! [A3: nat] :
% 2.52/1.56 ( ( member_nat @ A3 @ S )
% 2.52/1.56 => ( ( I2 @ ( J2 @ A3 ) )
% 2.52/1.56 = A3 ) )
% 2.52/1.56 => ( ! [A3: nat] :
% 2.52/1.56 ( ( member_nat @ A3 @ S )
% 2.52/1.56 => ( member_nat @ ( J2 @ A3 ) @ T ) )
% 2.52/1.56 => ( ! [B3: nat] :
% 2.52/1.56 ( ( member_nat @ B3 @ T )
% 2.52/1.56 => ( ( J2 @ ( I2 @ B3 ) )
% 2.52/1.56 = B3 ) )
% 2.52/1.56 => ( ! [B3: nat] :
% 2.52/1.56 ( ( member_nat @ B3 @ T )
% 2.52/1.56 => ( member_nat @ ( I2 @ B3 ) @ S ) )
% 2.52/1.56 => ( ! [A3: nat] :
% 2.52/1.56 ( ( member_nat @ A3 @ S )
% 2.52/1.56 => ( ( H @ ( J2 @ A3 ) )
% 2.52/1.56 = ( G @ A3 ) ) )
% 2.52/1.56 => ( ( groups1842438620at_nat @ G @ S )
% 2.52/1.56 = ( groups1842438620at_nat @ H @ T ) ) ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_48_sum_Oeq__general__inverses,axiom,
% 2.52/1.56 ! [B: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 2.52/1.56 ( ! [Y2: nat] :
% 2.52/1.56 ( ( member_nat @ Y2 @ B )
% 2.52/1.56 => ( ( member_nat @ ( K @ Y2 ) @ A )
% 2.52/1.56 & ( ( H @ ( K @ Y2 ) )
% 2.52/1.56 = Y2 ) ) )
% 2.52/1.56 => ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( ( member_nat @ ( H @ X2 ) @ B )
% 2.52/1.56 & ( ( K @ ( H @ X2 ) )
% 2.52/1.56 = X2 )
% 2.52/1.56 & ( ( Gamma @ ( H @ X2 ) )
% 2.52/1.56 = ( Phi @ X2 ) ) ) )
% 2.52/1.56 => ( ( groups1842438620at_nat @ Phi @ A )
% 2.52/1.56 = ( groups1842438620at_nat @ Gamma @ B ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_49_sum_Oeq__general,axiom,
% 2.52/1.56 ! [B: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
% 2.52/1.56 ( ! [Y2: nat] :
% 2.52/1.56 ( ( member_nat @ Y2 @ B )
% 2.52/1.56 => ? [X3: nat] :
% 2.52/1.56 ( ( member_nat @ X3 @ A )
% 2.52/1.56 & ( ( H @ X3 )
% 2.52/1.56 = Y2 )
% 2.52/1.56 & ! [Ya: nat] :
% 2.52/1.56 ( ( ( member_nat @ Ya @ A )
% 2.52/1.56 & ( ( H @ Ya )
% 2.52/1.56 = Y2 ) )
% 2.52/1.56 => ( Ya = X3 ) ) ) )
% 2.52/1.56 => ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( ( member_nat @ ( H @ X2 ) @ B )
% 2.52/1.56 & ( ( Gamma @ ( H @ X2 ) )
% 2.52/1.56 = ( Phi @ X2 ) ) ) )
% 2.52/1.56 => ( ( groups1842438620at_nat @ Phi @ A )
% 2.52/1.56 = ( groups1842438620at_nat @ Gamma @ B ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_50_sum_Ocong,axiom,
% 2.52/1.56 ! [A: set_nat,B: set_nat,G: nat > nat,H: nat > nat] :
% 2.52/1.56 ( ( A = B )
% 2.52/1.56 => ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ B )
% 2.52/1.56 => ( ( G @ X2 )
% 2.52/1.56 = ( H @ X2 ) ) )
% 2.52/1.56 => ( ( groups1842438620at_nat @ G @ A )
% 2.52/1.56 = ( groups1842438620at_nat @ H @ B ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_51_diff__commute,axiom,
% 2.52/1.56 ! [I2: nat,J2: nat,K: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
% 2.52/1.56 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_52_sum_Oswap,axiom,
% 2.52/1.56 ! [G: nat > nat > nat,B: set_nat,A: set_nat] :
% 2.52/1.56 ( ( groups1842438620at_nat
% 2.52/1.56 @ ^ [I: nat] : ( groups1842438620at_nat @ ( G @ I ) @ B )
% 2.52/1.56 @ A )
% 2.52/1.56 = ( groups1842438620at_nat
% 2.52/1.56 @ ^ [J: nat] :
% 2.52/1.56 ( groups1842438620at_nat
% 2.52/1.56 @ ^ [I: nat] : ( G @ I @ J )
% 2.52/1.56 @ A )
% 2.52/1.56 @ B ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_53_mult_Oright__neutral,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ A2 @ one_one_nat )
% 2.52/1.56 = A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_54_mult_Oleft__neutral,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ one_one_nat @ A2 )
% 2.52/1.56 = A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_55_fun__upd__upd,axiom,
% 2.52/1.56 ! [F: nat > nat,X: nat,Y: nat,Z: nat] :
% 2.52/1.56 ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ F @ X @ Y ) @ X @ Z )
% 2.52/1.56 = ( fun_upd_nat_nat @ F @ X @ Z ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_56_fun__upd__triv,axiom,
% 2.52/1.56 ! [F: nat > nat,X: nat] :
% 2.52/1.56 ( ( fun_upd_nat_nat @ F @ X @ ( F @ X ) )
% 2.52/1.56 = F ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_57_fun__upd__apply,axiom,
% 2.52/1.56 ( fun_upd_nat_nat
% 2.52/1.56 = ( ^ [F2: nat > nat,X4: nat,Y4: nat,Z2: nat] : ( if_nat @ ( Z2 = X4 ) @ Y4 @ ( F2 @ Z2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_58_order__refl,axiom,
% 2.52/1.56 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_59_order__refl,axiom,
% 2.52/1.56 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_60_lambda__one,axiom,
% 2.52/1.56 ( ( ^ [X4: nat] : X4 )
% 2.52/1.56 = ( times_times_nat @ one_one_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_61_partitions__def,axiom,
% 2.52/1.56 ( number1551313001itions
% 2.52/1.56 = ( ^ [P3: nat > nat,N: nat] :
% 2.52/1.56 ( ! [I: nat] :
% 2.52/1.56 ( ( ( P3 @ I )
% 2.52/1.56 != zero_zero_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ one_one_nat @ I )
% 2.52/1.56 & ( ord_less_eq_nat @ I @ N ) ) )
% 2.52/1.56 & ( ( groups1842438620at_nat
% 2.52/1.56 @ ^ [I: nat] : ( times_times_nat @ ( P3 @ I ) @ I )
% 2.52/1.56 @ ( set_ord_atMost_nat @ N ) )
% 2.52/1.56 = N ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_62_partitionsI,axiom,
% 2.52/1.56 ! [P2: nat > nat,N2: nat] :
% 2.52/1.56 ( ! [I3: nat] :
% 2.52/1.56 ( ( ( P2 @ I3 )
% 2.52/1.56 != zero_zero_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ one_one_nat @ I3 )
% 2.52/1.56 & ( ord_less_eq_nat @ I3 @ N2 ) ) )
% 2.52/1.56 => ( ( ( groups1842438620at_nat
% 2.52/1.56 @ ^ [I: nat] : ( times_times_nat @ ( P2 @ I ) @ I )
% 2.52/1.56 @ ( set_ord_atMost_nat @ N2 ) )
% 2.52/1.56 = N2 )
% 2.52/1.56 => ( number1551313001itions @ P2 @ N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_63_partitionsE,axiom,
% 2.52/1.56 ! [P2: nat > nat,N2: nat] :
% 2.52/1.56 ( ( number1551313001itions @ P2 @ N2 )
% 2.52/1.56 => ~ ( ! [I4: nat] :
% 2.52/1.56 ( ( ( P2 @ I4 )
% 2.52/1.56 != zero_zero_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ one_one_nat @ I4 )
% 2.52/1.56 & ( ord_less_eq_nat @ I4 @ N2 ) ) )
% 2.52/1.56 => ( ( groups1842438620at_nat
% 2.52/1.56 @ ^ [I: nat] : ( times_times_nat @ ( P2 @ I ) @ I )
% 2.52/1.56 @ ( set_ord_atMost_nat @ N2 ) )
% 2.52/1.56 != N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_64_le__zero__eq,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 2.52/1.56 = ( N2 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_65_mult__zero__left,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_66_mult__zero__right,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_67_mult__eq__0__iff,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ A2 @ B2 )
% 2.52/1.56 = zero_zero_nat )
% 2.52/1.56 = ( ( A2 = zero_zero_nat )
% 2.52/1.56 | ( B2 = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_68_mult__cancel__left,axiom,
% 2.52/1.56 ! [C: nat,A2: nat,B2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ C @ A2 )
% 2.52/1.56 = ( times_times_nat @ C @ B2 ) )
% 2.52/1.56 = ( ( C = zero_zero_nat )
% 2.52/1.56 | ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_69_mult__cancel__right,axiom,
% 2.52/1.56 ! [A2: nat,C: nat,B2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ A2 @ C )
% 2.52/1.56 = ( times_times_nat @ B2 @ C ) )
% 2.52/1.56 = ( ( C = zero_zero_nat )
% 2.52/1.56 | ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_70_zero__diff,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_71_diff__zero,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 2.52/1.56 = A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_72_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ A2 @ A2 )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_73_bot__nat__0_Oextremum,axiom,
% 2.52/1.56 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_74_le0,axiom,
% 2.52/1.56 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_75_diff__0__eq__0,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ zero_zero_nat @ N2 )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_76_diff__self__eq__0,axiom,
% 2.52/1.56 ! [M: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ M @ M )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_77_mult__is__0,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ M @ N2 )
% 2.52/1.56 = zero_zero_nat )
% 2.52/1.56 = ( ( M = zero_zero_nat )
% 2.52/1.56 | ( N2 = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_78_mult__0__right,axiom,
% 2.52/1.56 ! [M: nat] :
% 2.52/1.56 ( ( times_times_nat @ M @ zero_zero_nat )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_79_mult__cancel1,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ K @ M )
% 2.52/1.56 = ( times_times_nat @ K @ N2 ) )
% 2.52/1.56 = ( ( M = N2 )
% 2.52/1.56 | ( K = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_80_mult__cancel2,axiom,
% 2.52/1.56 ! [M: nat,K: nat,N2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ M @ K )
% 2.52/1.56 = ( times_times_nat @ N2 @ K ) )
% 2.52/1.56 = ( ( M = N2 )
% 2.52/1.56 | ( K = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_81_sum_Oneutral__const,axiom,
% 2.52/1.56 ! [A: set_nat] :
% 2.52/1.56 ( ( groups1842438620at_nat
% 2.52/1.56 @ ^ [Uu: nat] : zero_zero_nat
% 2.52/1.56 @ A )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_82_diff__is__0__eq_H,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ M @ N2 )
% 2.52/1.56 => ( ( minus_minus_nat @ M @ N2 )
% 2.52/1.56 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_83_diff__is__0__eq,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ( minus_minus_nat @ M @ N2 )
% 2.52/1.56 = zero_zero_nat )
% 2.52/1.56 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_84_gr0,axiom,
% 2.52/1.56 ord_less_nat @ zero_zero_nat @ ( p @ k ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_85_zero__reorient,axiom,
% 2.52/1.56 ! [X: nat] :
% 2.52/1.56 ( ( zero_zero_nat = X )
% 2.52/1.56 = ( X = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_86_zero__le,axiom,
% 2.52/1.56 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_87_mult__not__zero,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ A2 @ B2 )
% 2.52/1.56 != zero_zero_nat )
% 2.52/1.56 => ( ( A2 != zero_zero_nat )
% 2.52/1.56 & ( B2 != zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_88_divisors__zero,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ A2 @ B2 )
% 2.52/1.56 = zero_zero_nat )
% 2.52/1.56 => ( ( A2 = zero_zero_nat )
% 2.52/1.56 | ( B2 = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_89_no__zero__divisors,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( A2 != zero_zero_nat )
% 2.52/1.56 => ( ( B2 != zero_zero_nat )
% 2.52/1.56 => ( ( times_times_nat @ A2 @ B2 )
% 2.52/1.56 != zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_90_mult__left__cancel,axiom,
% 2.52/1.56 ! [C: nat,A2: nat,B2: nat] :
% 2.52/1.56 ( ( C != zero_zero_nat )
% 2.52/1.56 => ( ( ( times_times_nat @ C @ A2 )
% 2.52/1.56 = ( times_times_nat @ C @ B2 ) )
% 2.52/1.56 = ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_91_mult__right__cancel,axiom,
% 2.52/1.56 ! [C: nat,A2: nat,B2: nat] :
% 2.52/1.56 ( ( C != zero_zero_nat )
% 2.52/1.56 => ( ( ( times_times_nat @ A2 @ C )
% 2.52/1.56 = ( times_times_nat @ B2 @ C ) )
% 2.52/1.56 = ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_92_zero__neq__one,axiom,
% 2.52/1.56 zero_zero_nat != one_one_nat ).
% 2.52/1.56
% 2.52/1.56 thf(fact_93_lambda__zero,axiom,
% 2.52/1.56 ( ( ^ [H2: nat] : zero_zero_nat )
% 2.52/1.56 = ( times_times_nat @ zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_94_sum_Oneutral,axiom,
% 2.52/1.56 ! [A: set_nat,G: nat > nat] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( ( G @ X2 )
% 2.52/1.56 = zero_zero_nat ) )
% 2.52/1.56 => ( ( groups1842438620at_nat @ G @ A )
% 2.52/1.56 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_95_sum_Onot__neutral__contains__not__neutral,axiom,
% 2.52/1.56 ! [G: nat > nat,A: set_nat] :
% 2.52/1.56 ( ( ( groups1842438620at_nat @ G @ A )
% 2.52/1.56 != zero_zero_nat )
% 2.52/1.56 => ~ ! [A3: nat] :
% 2.52/1.56 ( ( member_nat @ A3 @ A )
% 2.52/1.56 => ( ( G @ A3 )
% 2.52/1.56 = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_96_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.52/1.56 => ( A2 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_97_bot__nat__0_Oextremum__unique,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.52/1.56 = ( A2 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_98_le__0__eq,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 2.52/1.56 = ( N2 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_99_less__eq__nat_Osimps_I1_J,axiom,
% 2.52/1.56 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_100_diffs0__imp__equal,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ( minus_minus_nat @ M @ N2 )
% 2.52/1.56 = zero_zero_nat )
% 2.52/1.56 => ( ( ( minus_minus_nat @ N2 @ M )
% 2.52/1.56 = zero_zero_nat )
% 2.52/1.56 => ( M = N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_101_minus__nat_Odiff__0,axiom,
% 2.52/1.56 ! [M: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 2.52/1.56 = M ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_102_mult__0,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( times_times_nat @ zero_zero_nat @ N2 )
% 2.52/1.56 = zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_103_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_104_mult__nonneg__nonpos2,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_105_mult__nonpos__nonneg,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_106_mult__nonneg__nonpos,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_107_mult__nonneg__nonneg,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 2.52/1.56 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_108_split__mult__neg__le,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
% 2.52/1.56 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 2.52/1.56 & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_109_mult__right__mono,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_110_mult__left__mono,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_111_mult__mono_H,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat,D: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ C @ D )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_112_mult__mono,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat,D: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ C @ D )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_113_not__one__le__zero,axiom,
% 2.52/1.56 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_114_zero__le__one,axiom,
% 2.52/1.56 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 2.52/1.56
% 2.52/1.56 thf(fact_115_partitions__zero,axiom,
% 2.52/1.56 ! [P2: nat > nat] :
% 2.52/1.56 ( ( number1551313001itions @ P2 @ zero_zero_nat )
% 2.52/1.56 = ( P2
% 2.52/1.56 = ( ^ [I: nat] : zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_116_mult__left__le,axiom,
% 2.52/1.56 ! [C: nat,A2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_117_mult__le__one,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
% 2.52/1.56 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_118_sum__nonpos,axiom,
% 2.52/1.56 ! [A: set_nat,F: nat > nat] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( groups1842438620at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_119_sum__nonneg,axiom,
% 2.52/1.56 ! [A: set_nat,F: nat > nat] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1842438620at_nat @ F @ A ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_120_mult__eq__self__implies__10,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( M
% 2.52/1.56 = ( times_times_nat @ M @ N2 ) )
% 2.52/1.56 => ( ( N2 = one_one_nat )
% 2.52/1.56 | ( M = zero_zero_nat ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_121_dual__order_Oantisym,axiom,
% 2.52/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_122_dual__order_Oantisym,axiom,
% 2.52/1.56 ! [B2: nat,A2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_123_dual__order_Oeq__iff,axiom,
% 2.52/1.56 ( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
% 2.52/1.56 = ( ^ [A4: set_nat,B4: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ B4 @ A4 )
% 2.52/1.56 & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_124_dual__order_Oeq__iff,axiom,
% 2.52/1.56 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 2.52/1.56 = ( ^ [A4: nat,B4: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ B4 @ A4 )
% 2.52/1.56 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_125_dual__order_Otrans,axiom,
% 2.52/1.56 ! [B2: set_nat,A2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ C @ B2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_126_dual__order_Otrans,axiom,
% 2.52/1.56 ! [B2: nat,A2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ C @ B2 )
% 2.52/1.56 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_127_linorder__wlog,axiom,
% 2.52/1.56 ! [P: nat > nat > $o,A2: nat,B2: nat] :
% 2.52/1.56 ( ! [A3: nat,B3: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.52/1.56 => ( P @ A3 @ B3 ) )
% 2.52/1.56 => ( ! [A3: nat,B3: nat] :
% 2.52/1.56 ( ( P @ B3 @ A3 )
% 2.52/1.56 => ( P @ A3 @ B3 ) )
% 2.52/1.56 => ( P @ A2 @ B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_128_dual__order_Orefl,axiom,
% 2.52/1.56 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_129_dual__order_Orefl,axiom,
% 2.52/1.56 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_130_order__trans,axiom,
% 2.52/1.56 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X @ Y )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 2.52/1.56 => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_131_order__trans,axiom,
% 2.52/1.56 ! [X: nat,Y: nat,Z: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X @ Y )
% 2.52/1.56 => ( ( ord_less_eq_nat @ Y @ Z )
% 2.52/1.56 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_132_order__class_Oorder_Oantisym,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 2.52/1.56 => ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_133_order__class_Oorder_Oantisym,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.52/1.56 => ( A2 = B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_134_ord__le__eq__trans,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( B2 = C )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_135_ord__le__eq__trans,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( B2 = C )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_136_ord__eq__le__trans,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( A2 = B2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_137_ord__eq__le__trans,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( A2 = B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_138_order__class_Oorder_Oeq__iff,axiom,
% 2.52/1.56 ( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
% 2.52/1.56 = ( ^ [A4: set_nat,B4: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 2.52/1.56 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_139_order__class_Oorder_Oeq__iff,axiom,
% 2.52/1.56 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 2.52/1.56 = ( ^ [A4: nat,B4: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A4 @ B4 )
% 2.52/1.56 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_140_antisym__conv,axiom,
% 2.52/1.56 ! [Y: set_nat,X: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ Y @ X )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ X @ Y )
% 2.52/1.56 = ( X = Y ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_141_antisym__conv,axiom,
% 2.52/1.56 ! [Y: nat,X: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ Y @ X )
% 2.52/1.56 => ( ( ord_less_eq_nat @ X @ Y )
% 2.52/1.56 = ( X = Y ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_142_le__cases3,axiom,
% 2.52/1.56 ! [X: nat,Y: nat,Z: nat] :
% 2.52/1.56 ( ( ( ord_less_eq_nat @ X @ Y )
% 2.52/1.56 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 2.52/1.56 => ( ( ( ord_less_eq_nat @ Y @ X )
% 2.52/1.56 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 2.52/1.56 => ( ( ( ord_less_eq_nat @ X @ Z )
% 2.52/1.56 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 2.52/1.56 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 2.52/1.56 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 2.52/1.56 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 2.52/1.56 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 2.52/1.56 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 2.52/1.56 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_143_order_Otrans,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_144_order_Otrans,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_145_le__cases,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ~ ( ord_less_eq_nat @ X @ Y )
% 2.52/1.56 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_146_eq__refl,axiom,
% 2.52/1.56 ! [X: set_nat,Y: set_nat] :
% 2.52/1.56 ( ( X = Y )
% 2.52/1.56 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_147_eq__refl,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( X = Y )
% 2.52/1.56 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_148_linear,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X @ Y )
% 2.52/1.56 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_149_antisym,axiom,
% 2.52/1.56 ! [X: set_nat,Y: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X @ Y )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ Y @ X )
% 2.52/1.56 => ( X = Y ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_150_antisym,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X @ Y )
% 2.52/1.56 => ( ( ord_less_eq_nat @ Y @ X )
% 2.52/1.56 => ( X = Y ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_151_eq__iff,axiom,
% 2.52/1.56 ( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
% 2.52/1.56 = ( ^ [X4: set_nat,Y4: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X4 @ Y4 )
% 2.52/1.56 & ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_152_eq__iff,axiom,
% 2.52/1.56 ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
% 2.52/1.56 = ( ^ [X4: nat,Y4: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 2.52/1.56 & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_153_ord__le__eq__subst,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ( F @ B2 )
% 2.52/1.56 = C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_154_ord__le__eq__subst,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ( F @ B2 )
% 2.52/1.56 = C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_155_ord__le__eq__subst,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ( F @ B2 )
% 2.52/1.56 = C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_156_ord__le__eq__subst,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ( F @ B2 )
% 2.52/1.56 = C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_157_ord__eq__le__subst,axiom,
% 2.52/1.56 ! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( A2
% 2.52/1.56 = ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_158_ord__eq__le__subst,axiom,
% 2.52/1.56 ! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( A2
% 2.52/1.56 = ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_159_ord__eq__le__subst,axiom,
% 2.52/1.56 ! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( A2
% 2.52/1.56 = ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_160_ord__eq__le__subst,axiom,
% 2.52/1.56 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( A2
% 2.52/1.56 = ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_161_order__subst2,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_162_order__subst2,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_163_order__subst2,axiom,
% 2.52/1.56 ! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_164_order__subst2,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_165_order__subst1,axiom,
% 2.52/1.56 ! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_166_order__subst1,axiom,
% 2.52/1.56 ! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_167_order__subst1,axiom,
% 2.52/1.56 ! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: set_nat,Y2: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_168_order__subst1,axiom,
% 2.52/1.56 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_eq_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_169_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
% 2.52/1.56 = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_170_mult_Oassoc,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
% 2.52/1.56 = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_171_mult_Ocommute,axiom,
% 2.52/1.56 ( times_times_nat
% 2.52/1.56 = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_172_mult_Oleft__commute,axiom,
% 2.52/1.56 ! [B2: nat,A2: nat,C: nat] :
% 2.52/1.56 ( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
% 2.52/1.56 = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_173_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 2.52/1.56 ! [A2: nat,C: nat,B2: nat] :
% 2.52/1.56 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
% 2.52/1.56 = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_174_one__reorient,axiom,
% 2.52/1.56 ! [X: nat] :
% 2.52/1.56 ( ( one_one_nat = X )
% 2.52/1.56 = ( X = one_one_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_175_fun__upd__idem__iff,axiom,
% 2.52/1.56 ! [F: nat > nat,X: nat,Y: nat] :
% 2.52/1.56 ( ( ( fun_upd_nat_nat @ F @ X @ Y )
% 2.52/1.56 = F )
% 2.52/1.56 = ( ( F @ X )
% 2.52/1.56 = Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_176_fun__upd__twist,axiom,
% 2.52/1.56 ! [A2: nat,C: nat,M: nat > nat,B2: nat,D: nat] :
% 2.52/1.56 ( ( A2 != C )
% 2.52/1.56 => ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ A2 @ B2 ) @ C @ D )
% 2.52/1.56 = ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_177_fun__upd__other,axiom,
% 2.52/1.56 ! [Z: nat,X: nat,F: nat > nat,Y: nat] :
% 2.52/1.56 ( ( Z != X )
% 2.52/1.56 => ( ( fun_upd_nat_nat @ F @ X @ Y @ Z )
% 2.52/1.56 = ( F @ Z ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_178_fun__upd__same,axiom,
% 2.52/1.56 ! [F: nat > nat,X: nat,Y: nat] :
% 2.52/1.56 ( ( fun_upd_nat_nat @ F @ X @ Y @ X )
% 2.52/1.56 = Y ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_179_fun__upd__idem,axiom,
% 2.52/1.56 ! [F: nat > nat,X: nat,Y: nat] :
% 2.52/1.56 ( ( ( F @ X )
% 2.52/1.56 = Y )
% 2.52/1.56 => ( ( fun_upd_nat_nat @ F @ X @ Y )
% 2.52/1.56 = F ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_180_fun__upd__eqD,axiom,
% 2.52/1.56 ! [F: nat > nat,X: nat,Y: nat,G: nat > nat,Z: nat] :
% 2.52/1.56 ( ( ( fun_upd_nat_nat @ F @ X @ Y )
% 2.52/1.56 = ( fun_upd_nat_nat @ G @ X @ Z ) )
% 2.52/1.56 => ( Y = Z ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_181_fun__upd__def,axiom,
% 2.52/1.56 ( fun_upd_nat_nat
% 2.52/1.56 = ( ^ [F2: nat > nat,A4: nat,B4: nat,X4: nat] : ( if_nat @ ( X4 = A4 ) @ B4 @ ( F2 @ X4 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_182_left__diff__distrib_H,axiom,
% 2.52/1.56 ! [B2: nat,C: nat,A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A2 )
% 2.52/1.56 = ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_183_right__diff__distrib_H,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C ) )
% 2.52/1.56 = ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_184_comm__monoid__mult__class_Omult__1,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ one_one_nat @ A2 )
% 2.52/1.56 = A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_185_mult_Ocomm__neutral,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( times_times_nat @ A2 @ one_one_nat )
% 2.52/1.56 = A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_186_subsetI,axiom,
% 2.52/1.56 ! [A: set_nat,B: set_nat] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ( member_nat @ X2 @ A )
% 2.52/1.56 => ( member_nat @ X2 @ B ) )
% 2.52/1.56 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_187_subset__antisym,axiom,
% 2.52/1.56 ! [A: set_nat,B: set_nat] :
% 2.52/1.56 ( ( ord_less_eq_set_nat @ A @ B )
% 2.52/1.56 => ( ( ord_less_eq_set_nat @ B @ A )
% 2.52/1.56 => ( A = B ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_188_partitions__remove1__bounds,axiom,
% 2.52/1.56 ! [P2: nat > nat,N2: nat,K: nat,I2: nat] :
% 2.52/1.56 ( ( number1551313001itions @ P2 @ N2 )
% 2.52/1.56 => ( ( ord_less_nat @ zero_zero_nat @ ( P2 @ K ) )
% 2.52/1.56 => ( ( ( fun_upd_nat_nat @ P2 @ K @ ( minus_minus_nat @ ( P2 @ K ) @ one_one_nat ) @ I2 )
% 2.52/1.56 != zero_zero_nat )
% 2.52/1.56 => ( ( ord_less_eq_nat @ one_one_nat @ I2 )
% 2.52/1.56 & ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_189_nat__mult__eq__cancel__disj,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ( times_times_nat @ K @ M )
% 2.52/1.56 = ( times_times_nat @ K @ N2 ) )
% 2.52/1.56 = ( ( K = zero_zero_nat )
% 2.52/1.56 | ( M = N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_190_le__numeral__extra_I4_J,axiom,
% 2.52/1.56 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 2.52/1.56
% 2.52/1.56 thf(fact_191_le__numeral__extra_I3_J,axiom,
% 2.52/1.56 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 2.52/1.56
% 2.52/1.56 thf(fact_192_not__gr__zero,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 2.52/1.56 = ( N2 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_193_bot__nat__0_Onot__eq__extremum,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ( ( A2 != zero_zero_nat )
% 2.52/1.56 = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_194_less__nat__zero__code,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_195_neq0__conv,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( N2 != zero_zero_nat )
% 2.52/1.56 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_196_zero__less__diff,axiom,
% 2.52/1.56 ! [N2: nat,M: nat] :
% 2.52/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
% 2.52/1.56 = ( ord_less_nat @ M @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_197_nat__mult__less__cancel__disj,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 2.52/1.56 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.52/1.56 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_198_mult__less__cancel2,axiom,
% 2.52/1.56 ! [M: nat,K: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 2.52/1.56 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.52/1.56 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_199_nat__0__less__mult__iff,axiom,
% 2.52/1.56 ! [M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
% 2.52/1.56 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 2.52/1.56 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_200_less__one,axiom,
% 2.52/1.56 ! [N2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ N2 @ one_one_nat )
% 2.52/1.56 = ( N2 = zero_zero_nat ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_201_nat__mult__le__cancel__disj,axiom,
% 2.52/1.56 ! [K: nat,M: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 2.52/1.56 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.52/1.56 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_202_mult__le__cancel2,axiom,
% 2.52/1.56 ! [M: nat,K: nat,N2: nat] :
% 2.52/1.56 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 2.52/1.56 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 2.52/1.56 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_203_dual__order_Ostrict__implies__not__eq,axiom,
% 2.52/1.56 ! [B2: nat,A2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ B2 @ A2 )
% 2.52/1.56 => ( A2 != B2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_204_order_Ostrict__implies__not__eq,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ( A2 != B2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_205_not__less__iff__gr__or__eq,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.52/1.56 = ( ( ord_less_nat @ Y @ X )
% 2.52/1.56 | ( X = Y ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_206_dual__order_Ostrict__trans,axiom,
% 2.52/1.56 ! [B2: nat,A2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_nat @ B2 @ A2 )
% 2.52/1.56 => ( ( ord_less_nat @ C @ B2 )
% 2.52/1.56 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_207_linorder__less__wlog,axiom,
% 2.52/1.56 ! [P: nat > nat > $o,A2: nat,B2: nat] :
% 2.52/1.56 ( ! [A3: nat,B3: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A3 @ B3 )
% 2.52/1.56 => ( P @ A3 @ B3 ) )
% 2.52/1.56 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 2.52/1.56 => ( ! [A3: nat,B3: nat] :
% 2.52/1.56 ( ( P @ B3 @ A3 )
% 2.52/1.56 => ( P @ A3 @ B3 ) )
% 2.52/1.56 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_208_exists__least__iff,axiom,
% 2.52/1.56 ( ( ^ [P4: nat > $o] :
% 2.52/1.56 ? [X5: nat] : ( P4 @ X5 ) )
% 2.52/1.56 = ( ^ [P5: nat > $o] :
% 2.52/1.56 ? [N: nat] :
% 2.52/1.56 ( ( P5 @ N )
% 2.52/1.56 & ! [M4: nat] :
% 2.52/1.56 ( ( ord_less_nat @ M4 @ N )
% 2.52/1.56 => ~ ( P5 @ M4 ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_209_less__imp__not__less,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_210_order_Ostrict__trans,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_nat @ B2 @ C )
% 2.52/1.56 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_211_dual__order_Oirrefl,axiom,
% 2.52/1.56 ! [A2: nat] :
% 2.52/1.56 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_212_linorder__cases,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ~ ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( ( X != Y )
% 2.52/1.56 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_213_less__imp__triv,axiom,
% 2.52/1.56 ! [X: nat,Y: nat,P: $o] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( ( ord_less_nat @ Y @ X )
% 2.52/1.56 => P ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_214_less__imp__not__eq2,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( Y != X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_215_antisym__conv3,axiom,
% 2.52/1.56 ! [Y: nat,X: nat] :
% 2.52/1.56 ( ~ ( ord_less_nat @ Y @ X )
% 2.52/1.56 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.52/1.56 = ( X = Y ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_216_less__induct,axiom,
% 2.52/1.56 ! [P: nat > $o,A2: nat] :
% 2.52/1.56 ( ! [X2: nat] :
% 2.52/1.56 ( ! [Y3: nat] :
% 2.52/1.56 ( ( ord_less_nat @ Y3 @ X2 )
% 2.52/1.56 => ( P @ Y3 ) )
% 2.52/1.56 => ( P @ X2 ) )
% 2.52/1.56 => ( P @ A2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_217_less__not__sym,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_218_less__imp__not__eq,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( X != Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_219_dual__order_Oasym,axiom,
% 2.52/1.56 ! [B2: nat,A2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ B2 @ A2 )
% 2.52/1.56 => ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_220_ord__less__eq__trans,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( B2 = C )
% 2.52/1.56 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_221_ord__eq__less__trans,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( A2 = B2 )
% 2.52/1.56 => ( ( ord_less_nat @ B2 @ C )
% 2.52/1.56 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_222_less__irrefl,axiom,
% 2.52/1.56 ! [X: nat] :
% 2.52/1.56 ~ ( ord_less_nat @ X @ X ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_223_less__linear,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 | ( X = Y )
% 2.52/1.56 | ( ord_less_nat @ Y @ X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_224_less__trans,axiom,
% 2.52/1.56 ! [X: nat,Y: nat,Z: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( ( ord_less_nat @ Y @ Z )
% 2.52/1.56 => ( ord_less_nat @ X @ Z ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_225_less__asym_H,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_226_less__asym,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_227_less__imp__neq,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( X != Y ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_228_order_Oasym,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_229_neq__iff,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( X != Y )
% 2.52/1.56 = ( ( ord_less_nat @ X @ Y )
% 2.52/1.56 | ( ord_less_nat @ Y @ X ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_230_neqE,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( X != Y )
% 2.52/1.56 => ( ~ ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_231_gt__ex,axiom,
% 2.52/1.56 ! [X: nat] :
% 2.52/1.56 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_232_order__less__subst2,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_233_order__less__subst1,axiom,
% 2.52/1.56 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_234_ord__less__eq__subst,axiom,
% 2.52/1.56 ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
% 2.52/1.56 ( ( ord_less_nat @ A2 @ B2 )
% 2.52/1.56 => ( ( ( F @ B2 )
% 2.52/1.56 = C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_235_ord__eq__less__subst,axiom,
% 2.52/1.56 ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
% 2.52/1.56 ( ( A2
% 2.52/1.56 = ( F @ B2 ) )
% 2.52/1.56 => ( ( ord_less_nat @ B2 @ C )
% 2.52/1.56 => ( ! [X2: nat,Y2: nat] :
% 2.52/1.56 ( ( ord_less_nat @ X2 @ Y2 )
% 2.52/1.56 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 2.52/1.56 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_236_less__numeral__extra_I3_J,axiom,
% 2.52/1.56 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_237_less__numeral__extra_I4_J,axiom,
% 2.52/1.56 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_238_linorder__neqE__nat,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( X != Y )
% 2.52/1.56 => ( ~ ( ord_less_nat @ X @ Y )
% 2.52/1.56 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_239_infinite__descent,axiom,
% 2.52/1.56 ! [P: nat > $o,N2: nat] :
% 2.52/1.56 ( ! [N3: nat] :
% 2.52/1.56 ( ~ ( P @ N3 )
% 2.52/1.56 => ? [M5: nat] :
% 2.52/1.56 ( ( ord_less_nat @ M5 @ N3 )
% 2.52/1.56 & ~ ( P @ M5 ) ) )
% 2.52/1.56 => ( P @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(fact_240_nat__less__induct,axiom,
% 2.52/1.56 ! [P: nat > $o,N2: nat] :
% 2.52/1.56 ( ! [N3: nat] :
% 2.52/1.56 ( ! [M5: nat] :
% 2.52/1.56 ( ( ord_less_nat @ M5 @ N3 )
% 2.52/1.56 => ( P @ M5 ) )
% 2.52/1.56 => ( P @ N3 ) )
% 2.52/1.56 => ( P @ N2 ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
% 2.52/1.56 ! [P: $o] :
% 2.52/1.56 ( ( P = $true )
% 2.52/1.56 | ( P = $false ) ) ).
% 2.52/1.56
% 2.52/1.56 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( if_nat @ $false @ X @ Y )
% 2.52/1.56 = Y ) ).
% 2.52/1.56
% 2.52/1.56 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 2.52/1.56 ! [X: nat,Y: nat] :
% 2.52/1.56 ( ( if_nat @ $true @ X @ Y )
% 2.52/1.56 = X ) ).
% 2.52/1.56
% 2.52/1.56 thf(conj_0,conjecture,
% 2.52/1.56 ( ( minus_minus_nat
% 2.52/1.56 @ ( groups1842438620at_nat
% 2.52/1.56 @ ^ [I: nat] : ( times_times_nat @ ( p @ I ) @ I )
% 2.52/1.56 @ ( set_ord_atMost_nat @ n ) )
% 2.52/1.56 @ k )
% 2.52/1.56 = ( minus_minus_nat @ n @ k ) ) ).
% 2.52/1.56 ------------------------
% 2.52/1.58 % (2366)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/183Mi)
% 2.52/1.58 % (2368)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/27Mi)
% 2.52/1.58 % (2371)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/275Mi)
% 2.52/1.59 % (2367)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/4Mi)
% 2.52/1.59 % (2369)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/2Mi)
% 2.52/1.59 % (2370)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/2Mi)
% 2.52/1.59 % (2372)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/18Mi)
% 2.52/1.59 % (2368)Instruction limit reached!
% 2.52/1.59 % (2368)------------------------------
% 2.52/1.59 % (2368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (2368)Termination reason: Unknown
% 2.52/1.59 % (2368)Termination phase: Preprocessing 1
% 2.52/1.59
% 2.52/1.59 % (2368)Memory used [KB]: 1535
% 2.52/1.59 % (2368)Time elapsed: 0.010 s
% 2.52/1.59 % (2368)Instructions burned: 28 (million)
% 2.52/1.59 % (2368)------------------------------
% 2.52/1.59 % (2368)------------------------------
% 2.52/1.59 % (2369)Instruction limit reached!
% 2.52/1.59 % (2369)------------------------------
% 2.52/1.59 % (2369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (2369)Termination reason: Unknown
% 2.52/1.59 % (2369)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (2370)Instruction limit reached!
% 2.52/1.59 % (2370)------------------------------
% 2.52/1.59 % (2370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (2370)Termination reason: Unknown
% 2.52/1.59 % (2370)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (2370)Memory used [KB]: 1279
% 2.52/1.59 % (2370)Time elapsed: 0.004 s
% 2.52/1.59 % (2370)Instructions burned: 3 (million)
% 2.52/1.59 % (2370)------------------------------
% 2.52/1.59 % (2370)------------------------------
% 2.52/1.59 % (2369)Memory used [KB]: 1279
% 2.52/1.59 % (2369)Time elapsed: 0.004 s
% 2.52/1.59 % (2369)Instructions burned: 3 (million)
% 2.52/1.59 % (2369)------------------------------
% 2.52/1.59 % (2369)------------------------------
% 2.52/1.59 % (2367)Instruction limit reached!
% 2.52/1.59 % (2367)------------------------------
% 2.52/1.59 % (2367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.52/1.59 % (2367)Termination reason: Unknown
% 2.52/1.59 % (2367)Termination phase: shuffling
% 2.52/1.59
% 2.52/1.59 % (2367)Memory used [KB]: 1279
% 2.52/1.59 % (2367)Time elapsed: 0.005 s
% 2.52/1.59 % (2367)Instructions burned: 5 (million)
% 2.52/1.59 % (2367)------------------------------
% 2.52/1.59 % (2367)------------------------------
% 2.62/1.60 % (2372)Instruction limit reached!
% 2.62/1.60 % (2372)------------------------------
% 2.62/1.60 % (2372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.60 % (2372)Termination reason: Unknown
% 2.62/1.60 % (2372)Termination phase: Property scanning
% 2.62/1.60
% 2.62/1.60 % (2372)Memory used [KB]: 1535
% 2.62/1.60 % (2372)Time elapsed: 0.013 s
% 2.62/1.60 % (2372)Instructions burned: 18 (million)
% 2.62/1.60 % (2372)------------------------------
% 2.62/1.60 % (2372)------------------------------
% 2.62/1.60 % (2375)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/15Mi)
% 2.62/1.61 % (2375)Instruction limit reached!
% 2.62/1.61 % (2375)------------------------------
% 2.62/1.61 % (2375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.61 % (2375)Termination reason: Unknown
% 2.62/1.61 % (2375)Termination phase: shuffling
% 2.62/1.61
% 2.62/1.61 % (2375)Memory used [KB]: 1535
% 2.62/1.61 % (2375)Time elapsed: 0.007 s
% 2.62/1.61 % (2375)Instructions burned: 15 (million)
% 2.62/1.61 % (2375)------------------------------
% 2.62/1.61 % (2375)------------------------------
% 2.62/1.61 % (2373)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/3Mi)
% 2.62/1.61 % (2374)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/37Mi)
% 2.62/1.61 % (2376)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/3Mi)
% 2.62/1.61 % (2373)Instruction limit reached!
% 2.62/1.61 % (2373)------------------------------
% 2.62/1.61 % (2373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.61 % (2373)Termination reason: Unknown
% 2.62/1.61 % (2373)Termination phase: shuffling
% 2.62/1.61
% 2.62/1.61 % (2373)Memory used [KB]: 1279
% 2.62/1.61 % (2373)Time elapsed: 0.004 s
% 2.62/1.61 % (2373)Instructions burned: 3 (million)
% 2.62/1.61 % (2373)------------------------------
% 2.62/1.61 % (2373)------------------------------
% 2.62/1.61 % (2376)Instruction limit reached!
% 2.62/1.61 % (2376)------------------------------
% 2.62/1.61 % (2376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.61 % (2376)Termination reason: Unknown
% 2.62/1.61 % (2376)Termination phase: shuffling
% 2.62/1.61
% 2.62/1.61 % (2376)Memory used [KB]: 1279
% 2.62/1.61 % (2376)Time elapsed: 0.004 s
% 2.62/1.61 % (2376)Instructions burned: 4 (million)
% 2.62/1.61 % (2376)------------------------------
% 2.62/1.61 % (2376)------------------------------
% 2.62/1.62 % (2378)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/7Mi)
% 2.62/1.62 % (2377)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_2249 for (2999ds/1041Mi)
% 2.62/1.62 % (2378)Instruction limit reached!
% 2.62/1.62 % (2378)------------------------------
% 2.62/1.62 % (2378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.62 % (2378)Termination reason: Unknown
% 2.62/1.62 % (2378)Termination phase: shuffling
% 2.62/1.62
% 2.62/1.62 % (2378)Memory used [KB]: 1407
% 2.62/1.62 % (2378)Time elapsed: 0.004 s
% 2.62/1.62 % (2378)Instructions burned: 7 (million)
% 2.62/1.62 % (2378)------------------------------
% 2.62/1.62 % (2378)------------------------------
% 2.62/1.63 % (2371)First to succeed.
% 2.62/1.63 % (2371)Refutation found. Thanks to Tanya!
% 2.62/1.63 % SZS status Theorem for DTF2THF_2249
% 2.62/1.63 % SZS output start Proof for DTF2THF_2249
% 2.62/1.63 thf(type_def_5, type, nat: $tType).
% 2.62/1.63 thf(type_def_7, type, set_nat: $tType).
% 2.62/1.63 thf(type_def_8, type, set_set_nat: $tType).
% 2.62/1.63 thf(func_def_0, type, set_set_nat: $tType).
% 2.62/1.63 thf(func_def_1, type, set_nat: $tType).
% 2.62/1.63 thf(func_def_2, type, nat: $tType).
% 2.62/1.63 thf(func_def_3, type, fun_upd_nat_nat: (nat > nat) > nat > nat > nat > nat).
% 2.62/1.63 thf(func_def_4, type, minus_minus_nat: nat > nat > nat).
% 2.62/1.63 thf(func_def_5, type, one_one_nat: nat).
% 2.62/1.63 thf(func_def_6, type, times_times_nat: nat > nat > nat).
% 2.62/1.63 thf(func_def_7, type, zero_zero_nat: nat).
% 2.62/1.63 thf(func_def_8, type, groups1842438620at_nat: (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_9, type, if_nat: $o > nat > nat > nat).
% 2.62/1.63 thf(func_def_10, type, number1551313001itions: (nat > nat) > nat > $o).
% 2.62/1.63 thf(func_def_11, type, ord_less_nat: nat > nat > $o).
% 2.62/1.63 thf(func_def_12, type, ord_less_eq_nat: nat > nat > $o).
% 2.62/1.63 thf(func_def_13, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 2.62/1.63 thf(func_def_14, type, ord_le1613022364et_nat: set_set_nat > set_set_nat > $o).
% 2.62/1.63 thf(func_def_15, type, collect_nat: (nat > $o) > set_nat).
% 2.62/1.63 thf(func_def_16, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 2.62/1.63 thf(func_def_17, type, set_ord_atMost_nat: nat > set_nat).
% 2.62/1.63 thf(func_def_18, type, set_or1086813439et_nat: set_nat > set_set_nat).
% 2.62/1.63 thf(func_def_19, type, member_nat: nat > set_nat > $o).
% 2.62/1.63 thf(func_def_20, type, member_set_nat: set_nat > set_set_nat > $o).
% 2.62/1.63 thf(func_def_21, type, k: nat).
% 2.62/1.63 thf(func_def_22, type, n: nat).
% 2.62/1.63 thf(func_def_23, type, p: nat > nat).
% 2.62/1.63 thf(func_def_38, type, sP0: nat > nat > nat > $o).
% 2.62/1.63 thf(func_def_39, type, sP1: nat > nat > nat > $o).
% 2.62/1.63 thf(func_def_40, type, sP2: nat > nat > nat > $o).
% 2.62/1.63 thf(func_def_41, type, sP3: set_nat > (nat > nat) > (nat > nat) > $o).
% 2.62/1.63 thf(func_def_42, type, sP4: set_nat > (nat > nat) > set_nat > $o).
% 2.62/1.63 thf(func_def_43, type, sK5: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_44, type, sK6: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_45, type, sK7: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_46, type, sK8: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_47, type, sK9: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_48, type, sK10: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_49, type, sK11: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_50, type, sK12: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_51, type, sK13: set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_52, type, sK14: set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_53, type, sK15: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_54, type, sK16: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_55, type, sK17: (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_56, type, sK18: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_57, type, sK19: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_58, type, sK20: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_59, type, sK21: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_60, type, sK22: set_nat > (nat > nat) > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_61, type, sK23: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_62, type, sK24: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_63, type, sK25: (nat > $o) > nat).
% 2.62/1.63 thf(func_def_64, type, sK26: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_65, type, sK27: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_66, type, sK28: (nat > nat) > set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_67, type, sK29: set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_68, type, sK30: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_69, type, sK31: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_70, type, sK32: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_71, type, sK33: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_72, type, sK34: (nat > nat) > nat > nat).
% 2.62/1.63 thf(func_def_73, type, sK35: (nat > nat) > set_nat > (nat > nat) > (nat > nat) > set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_74, type, sK36: set_nat > (nat > nat) > (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_75, type, sK37: (nat > $o) > nat).
% 2.62/1.63 thf(func_def_76, type, sK38: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_77, type, sK39: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_78, type, sK40: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_79, type, sK41: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_80, type, sK42: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_81, type, sK43: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_82, type, sK44: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_83, type, sK45: nat > nat).
% 2.62/1.63 thf(func_def_84, type, sK46: set_nat > (nat > nat) > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_85, type, sK47: set_nat > (nat > nat) > (nat > nat) > set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_86, type, sK48: set_nat > set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_87, type, sK49: nat > set_nat > set_nat > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_88, type, sK50: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_89, type, sK51: (set_nat > nat) > set_nat).
% 2.62/1.63 thf(func_def_90, type, sK52: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_91, type, sK53: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_92, type, sK54: nat > (nat > $o) > nat).
% 2.62/1.63 thf(func_def_93, type, sK55: (nat > $o) > nat).
% 2.62/1.63 thf(func_def_94, type, sK56: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_95, type, sK57: (set_nat > set_nat) > set_nat).
% 2.62/1.63 thf(func_def_96, type, sK58: set_nat > (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_97, type, sK59: set_nat > (nat > nat) > (nat > nat) > nat).
% 2.62/1.63 thf(func_def_98, type, sK60: (nat > nat) > (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_99, type, sK61: set_nat > (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_100, type, sK62: (nat > nat) > (nat > nat) > (nat > nat) > set_nat > nat).
% 2.62/1.63 thf(func_def_101, type, sK63: set_nat > set_nat > nat).
% 2.62/1.63 thf(func_def_102, type, sK64: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_103, type, sK65: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_104, type, sK66: (nat > $o) > nat).
% 2.62/1.63 thf(func_def_105, type, sK67: (nat > $o) > nat).
% 2.62/1.63 thf(func_def_106, type, sK68: nat > (nat > $o) > nat).
% 2.62/1.63 thf(func_def_107, type, sK69: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_108, type, sK70: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_109, type, sK71: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_110, type, sK72: (nat > set_nat) > nat).
% 2.62/1.63 thf(func_def_111, type, sK73: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_112, type, sK74: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_113, type, sK75: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_114, type, sK76: (nat > nat > $o) > nat).
% 2.62/1.63 thf(func_def_115, type, sK77: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_116, type, sK78: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_117, type, sK79: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_118, type, sK80: (nat > nat) > nat).
% 2.62/1.63 thf(func_def_120, type, ph82: !>[X0: $tType]:(X0)).
% 2.62/1.63 thf(f1539,plain,(
% 2.62/1.63 $false),
% 2.62/1.63 inference(subsumption_resolution,[status(thm)],[f1529,f1156])).
% 2.62/1.63 thf(f1156,plain,(
% 2.62/1.63 ((number1551313001itions @ p @ n) = $true)),
% 2.62/1.63 inference(cnf_transformation,[status(thm)],[f380])).
% 2.62/1.63 thf(f380,plain,(
% 2.62/1.63 ((number1551313001itions @ p @ n) = $true)),
% 2.62/1.63 inference(fool_elimination,[status(thm)],[f379])).
% 2.62/1.63 thf(f379,plain,(
% 2.62/1.63 (number1551313001itions @ p @ n)),
% 2.62/1.63 inference(rectify,[status(thm)],[f1])).
% 2.62/1.63 thf(f1,axiom,(
% 2.62/1.63 (number1551313001itions @ p @ n)),
% 2.62/1.63 file('/export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2THF_2249.p',fact_0_partitions)).
% 2.62/1.63 thf(f1529,plain,(
% 2.62/1.63 ((number1551313001itions @ p @ n) != $true)),
% 2.62/1.63 inference(trivial_inequality_removal,[status(thm)],[f1522])).
% 2.62/1.63 thf(f1522,plain,(
% 2.62/1.63 ((minus_minus_nat @ n @ k) != (minus_minus_nat @ n @ k)) | ((number1551313001itions @ p @ n) != $true)),
% 2.62/1.63 inference(superposition,[status(thm)],[f1314,f1387])).
% 2.62/1.63 thf(f1387,plain,(
% 2.62/1.63 ( ! [X0 : nat > nat,X1 : nat] : (((groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (X0 @ Y0) @ Y0)) @ (set_ord_atMost_nat @ X1)) = X1) | ((number1551313001itions @ X0 @ X1) != $true)) )),
% 2.62/1.63 inference(cnf_transformation,[status(thm)],[f798])).
% 2.62/1.63 thf(f798,plain,(
% 2.62/1.63 ! [X0 : nat > nat,X1 : nat] : (((number1551313001itions @ X0 @ X1) != $true) | (((groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (X0 @ Y0) @ Y0)) @ (set_ord_atMost_nat @ X1)) = X1) & ! [X2 : nat] : ((zero_zero_nat = (X0 @ X2)) | (((ord_less_eq_nat @ X2 @ X1) = $true) & ((ord_less_eq_nat @ one_one_nat @ X2) = $true)))))),
% 2.62/1.63 inference(ennf_transformation,[status(thm)],[f618])).
% 2.62/1.63 thf(f618,plain,(
% 2.62/1.63 ! [X0 : nat > nat,X1 : nat] : (((number1551313001itions @ X0 @ X1) = $true) => ~(! [X2 : nat] : ((zero_zero_nat != (X0 @ X2)) => (((ord_less_eq_nat @ X2 @ X1) = $true) & ((ord_less_eq_nat @ one_one_nat @ X2) = $true))) => ((groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (X0 @ Y0) @ Y0)) @ (set_ord_atMost_nat @ X1)) != X1)))),
% 2.62/1.63 inference(fool_elimination,[status(thm)],[f617])).
% 2.62/1.63 thf(f617,plain,(
% 2.62/1.63 ! [X0 : nat > nat,X1 : nat] : ((number1551313001itions @ X0 @ X1) => ~(! [X2 : nat] : ((zero_zero_nat != (X0 @ X2)) => ((ord_less_eq_nat @ one_one_nat @ X2) & (ord_less_eq_nat @ X2 @ X1))) => ((groups1842438620at_nat @ (^[X3 : nat] : (times_times_nat @ (X0 @ X3) @ X3)) @ (set_ord_atMost_nat @ X1)) != X1)))),
% 2.62/1.63 inference(rectify,[status(thm)],[f64])).
% 2.62/1.63 thf(f64,axiom,(
% 2.62/1.63 ! [X23 : nat > nat,X11 : nat] : ((number1551313001itions @ X23 @ X11) => ~(! [X45 : nat] : ((zero_zero_nat != (X23 @ X45)) => ((ord_less_eq_nat @ one_one_nat @ X45) & (ord_less_eq_nat @ X45 @ X11))) => ((groups1842438620at_nat @ (^[X2 : nat] : (times_times_nat @ (X23 @ X2) @ X2)) @ (set_ord_atMost_nat @ X11)) != X11)))),
% 2.62/1.63 file('/export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2THF_2249.p',fact_63_partitionsE)).
% 2.62/1.63 thf(f1314,plain,(
% 2.62/1.63 ((minus_minus_nat @ n @ k) != (minus_minus_nat @ (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ n)) @ k))),
% 2.62/1.63 inference(cnf_transformation,[status(thm)],[f644])).
% 2.62/1.63 thf(f644,plain,(
% 2.62/1.63 ((minus_minus_nat @ n @ k) != (minus_minus_nat @ (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ n)) @ k))),
% 2.62/1.63 inference(flattening,[status(thm)],[f309])).
% 2.62/1.63 thf(f309,plain,(
% 2.62/1.63 ~((minus_minus_nat @ n @ k) = (minus_minus_nat @ (groups1842438620at_nat @ (^[Y0 : nat]: (times_times_nat @ (p @ Y0) @ Y0)) @ (set_ord_atMost_nat @ n)) @ k))),
% 2.62/1.63 inference(fool_elimination,[status(thm)],[f308])).
% 2.62/1.63 thf(f308,plain,(
% 2.62/1.63 ~((minus_minus_nat @ (groups1842438620at_nat @ (^[X0 : nat] : (times_times_nat @ (p @ X0) @ X0)) @ (set_ord_atMost_nat @ n)) @ k) = (minus_minus_nat @ n @ k))),
% 2.62/1.63 inference(rectify,[status(thm)],[f246])).
% 2.62/1.63 thf(f246,negated_conjecture,(
% 2.62/1.63 ~((minus_minus_nat @ (groups1842438620at_nat @ (^[X2 : nat] : (times_times_nat @ (p @ X2) @ X2)) @ (set_ord_atMost_nat @ n)) @ k) = (minus_minus_nat @ n @ k))),
% 2.62/1.63 inference(negated_conjecture,[status(cth)],[f245])).
% 2.62/1.63 thf(f245,conjecture,(
% 2.62/1.63 ((minus_minus_nat @ (groups1842438620at_nat @ (^[X2 : nat] : (times_times_nat @ (p @ X2) @ X2)) @ (set_ord_atMost_nat @ n)) @ k) = (minus_minus_nat @ n @ k))),
% 2.62/1.63 file('/export/starexec/sandbox/tmp/tmp.YxszVcafJq/DTF2THF_2249.p',conj_0)).
% 2.62/1.63 % SZS output end Proof for DTF2THF_2249
% 2.62/1.63 % (2371)------------------------------
% 2.62/1.63 % (2371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.63 % (2371)Termination reason: Refutation
% 2.62/1.63
% 2.62/1.63 % (2371)Memory used [KB]: 6780
% 2.62/1.63 % (2371)Time elapsed: 0.043 s
% 2.62/1.63 % (2371)Instructions burned: 98 (million)
% 2.62/1.63 % (2371)------------------------------
% 2.62/1.63 % (2371)------------------------------
% 2.62/1.63 % (2365)Success in time 0.062 s
%------------------------------------------------------------------------------